This is a slightly more cunning approach. There ARE certain specific set of circumstances where heat alters the sound (and, later on, results in ie speaker damage). If you drive your speakers in the red for a prolonged listening session, you may hear the sound deteriorating. I would assume this is the basis for the „heat sink“ „theory“.
My belief is that the smartest and most successful marketing always relies on some fatcs. There's always at least 5% of truth in every BS marketing. The rest is just BS.
But it's still BS in the end. Because, as
@ski2xblack said “virtually all modern amps from reputable companies“ WILL NOT have overheating issues if used within their limits.
BS marketing is not spread only by those who have direct material interest. All of the tricked and fooled ones will push it simply because only other choice is to addmit they've been fooled. And believe me no one will opt for that. And all the placebo affected ones will push it because they really belieive they heard something.
Even your friends, and good ones at that, might give you false info simply because they honestly think they heard some night'n'day difference. It would be far less harmfull if there weren't a price tag attached to it.
OTOH power conditioners are 99% BS, the remainig 1% representing fautly power installations and that is something you don't fix by power conditioners.