Many entrance themes if they have lyrics, will end up with way.
WWE wrestler Triple H's entrance theme, "The Game", is largely a Villain Song, sung by Motorhead from Triple H's point-of-view. It's three-and-a-half minutes of gloating about how Trips is the baddest of the bad and controls the WWE with an iron fist. Lemmy even throws in an Evil Laugh or two along the way.
A subversion: "Stand Back" was sung by Vince McMahon with the original intention to be used as André the Giant's song (and was actually used as such for a promo video and in the NES Wrestlemania game), but wound up being used for McMahon himself, who wasn't actually a villain at the time. (His on-screen persona was a face-biased announcer, nothing more) That changed a few years later, when he got a new persona and a new theme:
I'd want to come out to Vince McMahon's "No Chance" song, just so I could perform the "I have complete and utter disdain for everyone in this building, and that's why I'm walking with an exaggerated swagger and frowning" routine while 20,000 people lustily boo me. That seems like more fun than humans should be allowed.
As much of a douche as The Miz is...his song fits the trope.
"I came to PLAY! There's a price to pay, time for you to get down on your knees and pray. Say GOODBYE to the good ole days, they're never coming back, watch your future fade.
The Mountie: Jacques Rougeau Jr.'s heel marching tune, "The Mountie." When Rougeau began teaming with Pierre Oulette as part of The Quebecers, the tune was changed to "We're Not the Mounties" but – aside from minor lyrical differences and the vocal track replaced with a duet – kept pretty much intact.
Combine an ominous bassline, an eerily distorted guitar, a dark piano melody and haunting vocals and you have Live In Fear/Broken Out in Love, the perfectly sinister track that accompanies The Wyatt Family in their promos and entrance.