After Guts has a breakdown when they have sex for the first time, she gives him one of these. Both cases become worse when you realize that both of the receivers of Casca's Hugs o' Healing may be badass, but are painfully starved for genuine affection and intimacy.
During the final battle of Mai-HiME, Mai does this to a Mind Control-influenced Mikoto...and it works. For those of you keeping score at home, that marks the second time Mai's saved/spared Mikoto's life (the first time came in the first episode when Mai rescued her from drowning).
Natsuki does the same thing to Shizuru, just one episode prior. Though it was really the kiss that had more effect, the hug was definitely part of it.
In Mai Otome, when Nina tries to run off to look for her father Sergay after the canceled survival exam, Shizuru stops her, slaps her to bring her to her senses, tells her they are doing everything they can to find him, and then follows with a Cooldown Hug.
The last episode of Revolutionary Girl Utena has a rather cruel subversion, since it looks like Anthy is doing this to Utena as she's getting PO-ed at Akio, but it turns out that Anthy just stabbed Utena in the back — literally, with a sword.
Possibly explained -though not properly used- in the Komi Naoshi oneshot, Koi No Kami Sama: if chaos and Bad Things are happening around but not to Subject A, then the safest place to be is wrapped around Subject A's shoulders. The story being what it is, the hug is key to stopping the Chaos and Bad Things.
Clare then does the same to Jeane when she's about to awake, complemented with some direct manipulation of Jeane's powers.
Raki attempts it again at the end of the series, when Clare has presumably gone too far beyond the point of no return. It doesn't work, since struggling with herself for a few moments Clare attacks him, but ends up killing a fellow Claymore who knocks Raki out of the way at the last moment. That does make her snap out of it.
That scene happens slightly differently in the manga, with an already badly wounded Jean grabbing Clare and aligning their energy, turning Clare back. However, she does get stabbed even more in the process, and she dies.
Naruto has Sakura do this to Sasuke in the Forest of Death as his Cursed Seal begins to act up. Much later, this is subverted when she tries to do the same to Naruto and it doesn't work.
And much later, Naruto himself gives one to a darker version of himself.
Mochizuki Oboro in the manga Psyren has a slight deviation of the this as his actual power. He can heal people by hugging them.
Hiryuu does this to Tatsuo at one point.
In InuYasha, Kagome does this to Inuyasha twice to revert him from his Superpowered Evil Side, once in the second movie and once in the last episode of the anime. The movie instance also involved a kiss.
Which is odd because he was baring his teeth at the time of the kiss.
Which is notably different from most of the examples because it's done by someone who didn't really have a close relationship to Ed. Normally a cooldown hug is used as an affirmation of the huggers true love for the hugee (be it platonic or romantic).
In the dubbed version and its subtitling, Ed can be heard (very softly) saying something along the lines of, "Mom... You're back," when hugged by Lt. Ross, thus confirming the emotional parental transference he experienced at that time which allowed the Cooldown Hug to work.
In Busou Renkin, after Kazuki tuns into a victor, Tokiko's hug is what turns him back into a human. It wasn't permanent though.
In the Galaxy Fraulein Yuna OVA, Ayako does this after convincing her older sister Gemmu to stop fighting. Subverted, as Gemmu hugs her sister lovingly, then rips out her power core, which is coincidentally located in her chest, making the scene horrifyingly symbolic. Also played straight two episodes earlier than this, albeit coupled with an Armor-Piercing Slap to get Ayako's attention.
In the manga Crimson Spell, spellbreaker Halvir at one point embraces cursed Prince Valdrigue to calm him down and prevent his Superpowered Evil Side from taking over. Of course, on numerous other occasions Havi allows (if not outright enables) Vald's curse to take over and then keeps the Superpowered Evil Side under control with sex.
Played straight in Soul Eater as Maka fires up her Black Blood to make herself as crazy as Crona, so that she can "synchronize" with him/her; connect psychically, then give him/her physical affection; even while Crona is trying to stab her. Since Maka hardens her Black Blood to deflect the hits, she's fine.
Another, example from Soul Eater: Marie hugs Stein to bring him back from Medusa's insanity.
Pokémon Ash has given one to Pikachu in Hoenn, and Chimchar in Sinnoh. Both actually bit him during the process, although not intentionally.
Monferno goes under the effect of Blaze again in DP 163, and it takes two tries for Ash to get it under control. After this it evolves into Infernape.
Resulting in the Count's death of course, but that's a mere detail.
In the manga chapter of Magical Girl Lyrical Nanoha that features the Lightning Squad, it's revealed that his time in the illegal research facility left Erio very emotionally unstable and distrustful of everyone. When she was about to adopt him, Fate does this to Erio during one of his bouts of anger after he burns her hands.
To Love-Ru Darkness has Rito delivering one to Nana when she is hiding in her animal reserve and crying about Mea's decision to break off their (sadly one-sided) "false friendship".
Also, Sumire's hug to a weeping Lucia in the second volume of the manga.
Classically subverted in the last few pages of Lone Wolf and Cub. After Itto is finally Killed Off for Real, Retsudo holds his arms out like he's ready to grab and disarm little Daigoro. It turns out that Retsudo just wanted a hug, but that doesn't stop Daigoro from running him through.
That's basically the primary combat technique of Rushuna in Grenadier. E.g. she defeats her Evil Twin Setsuna by hugging her, despite sustaining heavy wounds and falling unconscious. All Setsuna can do after seeing that is admit her defeat. That is also how Yajirou defeats Big Bad Kaizan Doushi... except for the small detail that by hugging him, he causes him to fall some fifty meters onto a solid surface. He wasn't taking any chances, was he?
In Ranma ½, Ranma could only be brought out of the trance of the Cat Fist by an old lady down the street. Since she's visible only in a flashback, the responsibility falls on Akane, who will throw herself into Ranma's way to grab him and calm him down. Fortunately for everyone involved, Cat-Fist Ranma utterly adores her, and gladly leaps onto her lap or into her arms whenever she calls him.
In Deadman Wonderland, Karako gives Nagi one of these when Nagi went berserk. And she gets run through by Genkaku for it.
Another successful usage occurred in its sequel, Umineko No Naku Koro Ni. Ange uses this in order to snap Battler out of his Heroic BSOD. Unfortunately, thanks to doing this, Ange herself dies. But hey, it worked, right?
A variation is used in The Melancholy of Haruhi Suzumiya: In the last episode of the first arc, Kyon kisses Haruhi in a closed space variant to stop her from destroying and remaking the world.
Witch Hunter Robin gives one to Haruto Sakaki in the episde "Smells Like the Wandering Spirit." Her willingness to be openly affectionate to a teammate is one of the things that sets her apart from her more uptight teammates.
Shinn does this to Stella several times during Gundam SEED Destiny, including a quasi-psychic one during the Destroy battle. The last one fails. Miserably
A variant occurs a couple of times in Junjou Romantica: when Misaki can't stop crying, Usagi kisses him, and the shock is enough to calm Misaki down; in one of the "Junjou Minimum" flashbacks, we see that he learned this trick as a child and used it on Hiroki when he was having a meltdown due to self-induced stress.
Canute calms Bjorn with strong words and a hug in Vinland Saga. This leads to tragedy when Atli of all people stabs Bjorn in the back.
Hisoka Kurosaki gives this to his partner Asato Tsuzuki in the Kyoto arc of Yami No Matsuei, in order to stop him from commiting suicide after a particularly devistating encounter with Muraki, pleading at Tsuzuki to live for him even if he doesn't have anything else.
Happens with Sena in Eyeshield 21. After beating the Bando Spiders, him and Suzuna bump into each other. Suzuna starts talking, but Sena just collapses in her arms, having exerted all his strength that game.
In Canaan, Minoru slowly walks up to Hakko when she suffers a HeroicBSOD after Santana dies and hugs her. Color-coded for your convenience. There's a subversion, though: Minoru does manage to calm her down enough to keep her from shooting everyone else... but it's not enough to bring her back from the Despair Event Horizon. Soon afterwards, the poor girl simply lets herself die in the collapsing labs, next to Santana's lifeless body.
The Violinist of Hameln: Flute gives these to Hamel on several occasions in the manga: The first time is just after The Reveal of his demon horn, when he went berserk and was only stopped by Flute running up and embracing him from behind. Later, Flute giving him these seems to be the only way to turn him back to normal when Hamel's demon blood takes over, though the first time he actually put his fist through her chest before succumbing to the hug.
In the manga Wild Adapter, Kubota gives Tokito one of these in vol 5 when Tokito goes completely berserk on hearing what might be his real name. All the creepier because Tokito breaks his arm for touching him, but Kubota doesn't even flinch, and continues to soothe him. It scares Shouta, who caused it inadvertently, completely out of his mind.
In Bleach, Orihime stops her Hollow-fied brother from further rampage with a hug and gentle words.
At the end of the Fruits Basket anime (and volume six of the manga) Tohru encounters Kyo in his true form, clutches tightly to his arm and tells him that she is scared of how he looks now, but that she is still his friend and wants to be together with him. Kyo calms down and changes back to normal, explaining his mother always claimed to love him but was terrified of his true form and was always checking to make sure the beads were in place. He tells Tohru "You dont have to love everything" and then he hugs her, changing him back to his regular cat form.
Kyo gives one to Tohru in volume nineteen when she's panicking about being a terrible person for pretending to be as much like as her dead father as possible so her mother wouldn't leave (after being in shock after his death).
Subverted in volume twenty when Kureno hugs Akito and she stabs him.
This happens early in the Chrono Crusade anime, in the form of a Cooldown Tackle, when Chrono goes berserk when he thinks Rosette has been injured, and starts to forcibly release the seal on his demonic powers.
In Rave Master, Elie hugged a crying Haru who is grieving for the lost of his father.
Played straight in Eureka Seven, with Renton hugging and comforting Eureka in episode 35 when she is in depression.
Lloyd also hugs Colette to get her angel powers to shut off when they're going a little haywire.
In Serial Experiments Lain, Arisu panics to the point of slapping Lain hard enough to draw blood while thrashing about. Arisu had her arm raised for a second blow when Lain finally manages to get her into a hug, and well... it doesn't exactly fix her, but she does calm down.
In Samurai Deeper Kyo, Akira does this to a girl whose power is to release deadly corrosive spores into the air (her first ever hug, obviously), it startled her enough to manage to stop the flow.
In Ookami Kakushi, Kaori does this to Sakaki after he has a bit of a freakout. And then it turns out she's hugging him to drag him off a cliff.
In Angel Beats!, Otonashi does this to Naoi, who's extremely disillusioned with his life and wants to make everyone vanish by using his hypnosis powers to give them false memories*
. It works, and Naoi remains a... close friend of Otonashi for the rest of the series.
Last Order gives one to Accelerator in volume 15 of the To Aru Majutsu no Index when After beating the #2 esper since the #2 (Kakine) tried use civilians and Last Order as hostages to beat Accelerator, Accelerator goes to finish Kakine off, only for Yomikawa to stop him. Kakine stabs Yomikawa and Accelerator almost goes into an Unstoppable rage, tearing Kakine's arm off an is about to attack the anti-skill soldier nearby when Last Order stops him with a hug
Sylphiel gives one to Lina in Slayers NEXT, when she freaks out over thinking she's killed a little girl. Said kid turns out to be an Undead Child.
Sheryl does this to Ranka in Macross Frontier when she gets depressed of Alto is with the former.
In Blue Exorcist, The crippling self-doubt from which Rin suffers in Chapter 26 is ended when he receives a Cooldown Hug from Shiemi. What makes this even more heartwarming is that they were on fire at the moment, and Shiemi is an ex-Hikikomori who is still kind of a Shrinking Violet — she did so in order to prove her trust that he would not burn her with said flames.
In Clannad, Nagisa does this to Tomoya when his father goes to jail and Tomoya loses his chances of landing a very well paying job because of it. Tomoya snaps and begins pummelling the wall with his bare fists; Nagisa manages to embrace him and pin him to the wall to get him to stop. Once he's had a few seconds to breathe, he then asks her to marry him. She says yes.
Caesar to Nakaba in chapter 5 of Reimei No Arcana when she's freaking out over blood.
Cruelly averted in Gasaraki: Daizaburo is about to give one to Misuzu, to dissuade her from using her powers as a Kai — and right then, his eldest son Kazukiyo shoots him in the back, killing him.
In the Street Fighter IV: The Ties that Bind OVA, Sakura Kasugano gives one to Ryu when he falls under the influence of his Superpowered Evil Side, the infamous Evil Ryu. It's this close to not work since Ryu actually raises his fist to hit her, but he stops at the last moment and then slumps to his knees, with Sakura still hugging him.
In Sakura Gari, Souma gives one to Masataka after the torture and rape deal with Katsuragi. Somewhat subverted since Masataka at first rejects Souma's touch (understandably) and it's only after Souma promises to not go beyond and keeps his word that Masataka allows him to hug him close.
There is a tremor in her voice. A whisper in it that he can hear.
"Shinji, do you respect me?"
"I do," he says, "I wouldn't be here if I didn't."
"Good," she says with a shaky smile, "Because I'm going to completely f___ing lose it right now, okay?"
He pulls her to him as she begins crying. Her face buries in his shirt as she begins sobbing, screaming, her entire body shaking as he presses his lips to the top of her head. He waits, listening. Listening to the sounds of her whispers and hitched voice, listening to her as her tears wet his shirt. Arms wrapped around her, it takes hours for her to stop. Hours of her screaming, of her crying, of her whimpering in his arms.
Just so many in The Second Try... Usually is Shinji the one mades the embracing. Funny how he has become the most emotionally stable after the Third Impact.
The FIRST scene -that spawned the whole fic- consisted of Shinji embracing Asuka to cheer her up.
Shinji hugging Asuka in "Love" after their Freak Out.
Shinji hugging Asuka several times in "Repeat" because she was feeling awful.
Shinji sitting down with Asuka and embracing her after her Mind Rape (and Asuka stating who knows how much could have changed if he would have done that the first time)
A Growing Affection: Not his intent, but Naruto inadvertently gives one to Hinata in the first chapter. She intentionally returns the favor a few times later on.
Played with in X-Men: The Last Stand as Wolverine fights his way to Jean. Unusually for this trope, he's only able to calm her down momentarily, and takes that moment to stab her, though at Jean's request.
A drunken and remorseful Loki tries this on his insane partner-in-crime Bartleby at the climax of Dogma and gets a shiv in his kidney for his troubles.
Also happens between Bartleby and God right before the latter blows him up with her voice.
In the obscure and forgettable animated movie We're Back! A Dinosaur's Story, the dinosaurs have been brainwashed and turned evil, and the kid protagonists are able to change them back just by hugging them.
Done almost literally in the first Hellboy animated movie when Liz Sherman goes a little berserk in the first part of the movie with her pyrokinetic abilities, only stopping when Hellboy comes up close and tells her she can stop.
In Eastern Promises, Nikolai manages a borderline example on Kirill, who is, at the time, having something of a mental breakdown. Not least because he's been ordered to kill his infant half-sister. It's both a genuine cooldown attempt, as Nikolai seems to be the only one who can really deal with Kirill's mood swings, and a move to get the duffel bag the baby's in away from him.
In Tropic Thunder, just when it seems like Kirk Lazarus and Alpa Chino are going to get into a brawl, Lazarus hugs Chino and recites the lyrics to the theme song of The Jeffersons. Chino calls him on it, but the situation is still defused.
Though that may be less an example of the hug working and more that they had more pressing problems (namely, they just found out that they're stuck somewhere between Vietnam and Laos, Tugg Speedman has been captured by actual drug dealers, and Jeff Portnoy was starting to show serious signs of withdrawal) to deal with than N-Word Privileges.
Played straight and effectively in Mona Lisa Smile between Giselle (Maggie Gyllenhaal) and Betty (Kirsten Dunst).
Charles Xavier does this to Erik (Magneto) to save his life in X-Men: First Class when he almost drowns himself trying to lift Shaw's sub for the first time. Also comes with liberal applications of telepathic persuasion since they are both under water at the time.
In The Road Warrior, Humungus gives his lieutenant Wez one of these after Wez's bike buddy (and possibly lover) The Golden Youth is killed by the Feral Kid, causing Wez to go berserk at the refinery people, but Humungus basically locks his neck in his huge biceps, causing Wez to pass out before he can go on a suicidal attack.
Dragonlance has an example of this in The Soulforge, where Caramon does this to keep Raistlin from setting on fire/vaporizing/magic-ing the cultist woman who's denouncing him as a demon. Because this is still before Raistlin completely crossed the Moral Event Horizon, the cool-down hug works.
In The Elenium, Kalten does this to Sephrenia, who has blown her stack (as Sephrenia is a victim of racism all her life, Kalten recognizes her own racism towards the Delphae). It starts with him performing "Elene Magic" on the door by kicking it in, then giving her a hug. She can beat on him all she likes, but Kalten's body is about as hard as his head.
This is essentially how Xander "defeats" Big Bad Willow at the end of season six.
Giles and Buffy share a mutual hug at the end of "Passions", with Giles weeping over his dead lover Jenny Calender, and Buffy crying for him not to throw away his life in a Roaring Rampage of Revenge as "I can't do this alone."
Happens between Faith and Angel in "Five by Five". After spending the entire episode trying to get Angel to fight her, Faith breaks down crying, begging Angel to kill her. He simply hugs her until she calms down in the next episode.
Again with Faith and Angel, in "Sanctuary". Faith kills an assassin sent after her by Wolfram and Hart, freaking out when she sees the blood on her hands. Angel hugs her to calm her down, right as Buffy walks in the door.
In Power Rangers Zeo, during the I Know You Are In There Somewhere Fight between the Zeo Rangers and a brainwashed Tommy, Katherine demorphs to try and get him to stop fighting them. He does freeze, and she uses that opportunity to hug him, and hold on to him even when he tries to resist, eventually breaking the spell.
In Power Rangers Time Force, Jen temporarily gets brainwashed and beats the crap out of Wes. He manages to grap her into a hug and refuses to let her go, despite her struggling, until she snaps out of it.
In one episode of Star Trek ("The Changeling"), Kirk gives one of these to Spock after the latter tried to mind meld with a robot.
Near the end arcs of Kamen Rider Kiva, Maya tries to calm down her son, Taiga, with one of these. This is one of the "hugger gets stabbed" subversions, though she doesn't die.
Kamen Rider OOO needs one of these the first time he uses PuToTyranno form. After wiping out both of the monster he was fighting, he starts whaling on Kamen Rider Birth instead. When Hina tries to stop him, he almost hits her, but freezes at the last second. She takes the chance to grab him, which fully snaps him out of it.
Grey's Anatomy: The Asperger-autistic surgeon Virginia Dixon suffers from a overload after being congratulated too much by her colleagues. She then proceeds to actually ask two of the other women to give her a Cooldown Hug, even giving a scientific explanation for why this kind of hug actually works in Real Life.
The X-Files. Scully gives a big one to Mulder after his mother commits suicide.
In the series 4 premiere of The Inspector Lynley Mysteries, Barbara Havers is held hostage at gunpoint - on her first case back after taking a bullet to the stomach in the series 3 finale. She's terrified out of her skin, but largely keeps in control, until she is rescued - at which point she whales the tar out of the hostage-taker until Lynley manages to pull her off. She tries to hit him until it sinks in who's holding her - and then she bursts into tears and cries herself out against Lynley's chest.
Done literally on Heroes when Tracy, in the middle of a panic attack, turns Claire into ice when Claire attempts to calm her down. Poor woman then accidentally snaps off Claire's frozen foot and her panic attack worsens until a newly-healed Claire reassures Tracy that she's fine and the panic attacks subside.
In a first-season episode of Night Court, Judge Harry does this to calm down an angry teen-age runaway played by Michael J. Fox.
There's a game on Whose Line Is It Anyway? called Two Line Vocabulary, and in one playing, Ryan Stiles had the line "Somebody needs a hug." He repeats this consistently during the game, resulting in Colin hugging Brad when the latter would become hysterical (until Colin, as is normal for his character in the game, loses his temper and refuses hugs).
Done hilariously in the 2011 Christmas episode of the Panel Game8 Out of 10 Cats, when the panel featured Jedward and Greg Davies. After about twenty minutes of inane blathering from Jedward, Greg Davies breaks and starts mimicking Jedward's semi-surreal speech in frustration, descending into pure incoherent gibberish. He gets a consoling hug from team captain Jon Richardson.
Interestingly enough, the bearhug maneuver is a literal Cooldown Hug. It is one of a list of maneuvers known as a "rest-hold," which looks painful or dangerous, but is actually designed to let the two wrestlers catch their breath and cool down in between bigger power moves
Can also be used by a heel manager and his client to get heat from an unsympathetic crowd.
Solar Exalted have a Charm that allows them to enter an Unstoppable Rage. The mental costs for prematurely exiting the stage are significantly reduced if the Solar is restrained by a friend or loved one.
In the 9th chapter of Quintessence - The Blighted Venom, Serai was yelling to Reivier about all the trauma she'd received from being imprisoned and force-fed poison all because of Lunair, and Reivier gave her a sudden hug to shut her up. I mean, calm her down.
In the ending of Disgaea 2, Adell hugs Rozalin to snap her back to her senses when her memories and persona of her previous life as Overlord Zenon are restored to prevent the possible slaughter that would have ensued because of it.
Also used in Disgaea 3, where Raspberyl gives Mao a cooldown hug to stop him from totally snapping and turning into an Eldritch Abomination. It's subverted in the Bad Ending when the Big Bad, realizing what she's doing, kills her and sends Mao even farther off the deep end.
Nina does this to Ryu in Breath of Fire 4 after Ryu goes berserk and messily slaughters some Empire soldiers, wrecking an abandoned town in the process.
Kohak gives one to Shing in Tales Of Hearts, ending his thitherto escalating Heroic BSOD. She later gives another one to Chalcedny when he's getting frustrated that the operation to turn his Soma into a Cool Ship isn't panning out, making him pay attention to the wishes of his dying comrades, which kickstarts the transformation. (Nothing less from a game where all the Applied Phlebotinum is based on the Power of Friendship.)
A beautiful version of this occurs in Kingdom Hearts, after Sora has become a Heartless Kairi saves him from the darkness by hugging him and telling him to come back to her.
Marona does this in Phantom Bravewhenever Sprout is in danger of succumbing to the darkness and losing control. The last time she does it, it nearly fails, but gives him enough time for a Heroic Sacrifice to prevent his full power being used for the revival of Sulphur.
The Wii version of A Boy and His Blob has a button for this. It was going to be a gameplay mechanic but it was scrapped, so it has no use other than being unbelievably adorable.
In Mass Effect 2, Shepard has the option of giving one of these to Tali after she discovers her father's corpse during her loyalty mission.
Shepard can also hug Liara in the DLC Lair of the Shadow Broker, after she's similarly overwhelmed by the resolution of a years-long revenge quest.
Persona 3 has a memorable one in the trip to Yakushima Island, where Yukari Takeba has a breakdown and storms off after the party is shown a video implicating her father in the creation of the shadows they're fighting. The MC is sent off to calm her down and bring her back, and giving her one of these is one option in the ensuing conversation.
Her social link has another instance as well, where the MC gets the option to give her a hug after beating up a bunch of thugs that were harassing her. This one's a subversion, though - she'll be pissed if you select the option to hug her, since she'll see it as being condescending.
Rena tries to do this Keiichi during the "Spirited Away by the Demon" Chapter in Higurashi no Naku Koro ni even as Keiichi readies to bash her head in. He was too far gone for the hug to work but it does play in important role later on as a memory that causes Keiichi to remember the past events. It does work in the "Atonement Chapter" except that their roles are switched.
Persona 4 gives you the option to give Yosuke a hug during his Social Link. And it's around the waist, too.
Also Rin does this to Sakura who is currently in her evil mode. In the anime Rin runs up to Sakura as she's beginning a Super Power Meltdown. Slightly subverted as Sakura is shown to have stabbed her, though Rin appears to just walk it off.
In Katawa Shoujo, Hisao gives one to Rin during her first H-scene.
In Earthbound webcomic The Chosen Four, Ness gets lost in Dungeon Man and grows frustrated and angry to the point of he is bursting with psychic energy and he scares minions away with only glaring at them. Then his friends find him, Paula hugs him... and he suddenly feels completely cheered up. He even forgets what he was upset about.
Word Of God confirmed that is a hint of Giygas is trying influence him by fueling his negative emotions, but his friends are keeping him sane.
Rachel from Tower of God nearly has an emotional breakdown when she remembers Baam's Heroic Sacrifice, but his best friend Koon is there to hug her and cheer her up. It would be so touching if you didn't know that Rachel herself killed Baam and just made up that story, crying crocodiles tears and living off of Koon's kindness. The best part of it; Koon knows. He's just keeping Rachel around to get to her co-conspirators, the people he thinks are ultimately responsible for planning Baam's death. Not that Baam is dead, but that's an entirely different matter.
In Gunnerkrigg Court, Chapter 16, a spectacular hug is used to both figuratively and literally cool down the huggee.
Survival of the Fittest's Maxwell Crowe does this to a suicidal Maria Graham after chucking the broken glass shard she was going to do herself in with. This not only functions as a cooldown hug, but also prevents her from just pulling on her explosive collar and killing herself that way.
Subverted in the Whateley Universe: in "Ayla and the Mad Scientist", Team Kimba is running Dark Phoenix simulations in the school holographic simulator. When it's time for Tennyo to play Dark Tennyo, Generator insists on trying a Cooldown Hug first. She gets disintegrated for her efforts but it gives her teammates a chance to put four tranquilizer darts in Tennyo's backside.
In Worm, Chapter 14.8, when Skitter is trapped in the agnosia fog and deeply paranoid about trusting anyone, Grue — the teammate she has had a major crush on for a while, now — gives her one. Wait, did we say "Grue"? We meant Jack Slash — the Big Bad of the entire plot arc.
Kitty Pryde succeeds in calming down Wolverine with this technique in X-Men: Evolution.
She does this several times, actually. She's basically the walking off switch to his Berserk Button (ironically, her endangerment is usually the berserk button itself).
Which makes sense, actually. Since she's usually the one in danger, her hugging Aang basically tells his subconscious that she's all right, unharmed enough to embrace him, and he doesn't need to go berserk anymore.
In "The Desert," it was Appa's endangerment.
The first time it happened was in the third episode of the whole series: "The Southern Air Temple." It can be considered foreshadowing that they would become a Canon Couple.
Though Katara didn't hug him out of the Avatar State in "The Southern Air Temple." She just talked him out of it, and then when he came back to his senses she hugged him.
In Terra's premiere episode in Teen Titans, Beast Boy brings Terra back down to earth this way. Unfortunately, it's only a temporary solution...
In the Season 2 finale of Ben 10 Alien Force, Gwen transforms into a full Anodite about halfway through. She's certain that she could use her powers to destroy the threat right then and there, but Kevin, rightly fearing that she'd lose her humanity as a result, brings her back to earth with one of these.
In Codename Kids Next Door, a Friendly Friend Rainbow Monkey Goodbye Hug is the only way to calm down a giant rampaging rainbow monkey. Numbuh Four has to administer one after he pisses one off and it starts destroying the tree house.
The Cooldown Hug is a bit of Truth in Television as hugging has been proven to have health benefits like reducing blood pressure.
A lot of acting classes will set something up like this as a way of calming down after an emotional scene and reminding yourself that the person shouting at you a minute ago is actually your friend.
Not just a trope for humans: other social primates play this one straight, to bond with and calm one another after a conflict in the troop. Awww.....