If you pot the flop, I think you have to call a push.

How much have you been raising? Does villain respect your range? Could villain only be holding QQ, KK, AA in your eyes? What about JJ or suited AK?

vs. non heart AA you are a 57/42 dog
vs. heart AA you are 93/7
vs. QQ you are 66/33

His check looks trappy. I doubt non-heart AA checks here, so either he has a monster or he's holding JJ. I think betting here would only get you into trouble and fold worse hands, so I check and re-evaluate turn. But I side with monster here, due to his check. Proceed very cautiously, I think he needs a strong hand to be able to check that flop.