Alien Geometries: The temple was surely designed by an Eldritch Abomination. Or by M.C. Escher. Or by a terrain randomizer that doesn't keep track of where you have been, so that it may happily let you take seven quick 90 degree turns to the right in a row and come to a new location each time.
Or for Fridge Horror, the idol cursed you so that you can never leave, and you will be stuck there forever no matter how careful you are or how fast you run.
Camera Screw: When the minecart is running down curves in the sequel, the camera often turns at an angle that doesn't let you see the incoming traps.
Captain Ersatz: While you don't get to play as the world's most famous explorer, Indiana Jones, you do get to play as the world's second most famous explorer... Montana Smith. And yes, he wears the same iconic hat.
Continuing Is Painful: The price of continues doubles every time you use them. One upgrade, when maxed out, keeps the price at one for some tries, but after it's over it'll begin escalating from 3 to 11 and so on, until continuing won't be worth the cost.
Death as Comedy: Your deaths are often played for laughs. Whenever you die, you get a laconic text message. For example, when you die from falling into a pit, this message may be "I knew I should have learned how to fly" or "temple slippery when wet". These two messages have a double irony, since you can use "wings of resurrection" and since you can slide without problem over even the roughest surfaces.
Death Is Cheap: it only takes one click to get back on your feet after death. You have to start over as far as running distance is concerned, but you get to keep all the coins you collected on your previous runs. You can also buy the ability to resurrect yourself, so you can keep your running distance as well.
Follow the Leader: There are many clones of Temple Run that have spawned in the App Store from time to time, such as Subway Surfers and Run N Gun. Opinions will vary on if they're better or worse.
Similarly, Temple Run was arguably what started the big boom in "Freemium" games, causing many other iOS titles to adapt a similar currency system where you win money or other prizes based on how you preform in game and can spend real-life money to increase that amount to ridiculous numbers. Even the Angry Birds series could not avoid this.
Token Minority: The story seems to be taking place in Asia (judging from the architecture and rainforest), but all characters but one is western. The one non-western character is Asian. Her name? "Karma Lee".
Oh, and the western characters are either white or African-American. The white characters are explorers and escape artists and similar stuff that fit the setting, while the black characters have concepts such as cop or football star - making one wonder how he ended up in an ancient temple in the first place.
Visible Invisibility: Type 4. Interestingly, the monsters can still see you while you're invisible, but it does protect you against trees and pitfalls. One can only guess that the invisibility makes you intangible, and that the monsters are quite supernatural. (Well, they do have skulls for heads.)