Though you pose an interesting point, I feel compelled to point out that energy doesn't necessarily push matter apart. If it can push pieces apart, can it not also push them back together? And, we all know gravity can pull things together, but it also pulls things apart. And, when you think about it, energy can do BOTH. It can either repel or attract.

Rubidium has been observed at extremely low temperatures to repel other rubidium atoms, when it normally attracts them). Cool it down further, and it reverts to attraction again (or maybe it was the other way around). The same energy that attracts does just the opposite under certain circumstances, and gravity doesn't appear to have anything to do with this phenomenon.

So, given your explanation of why gravity is the "antienergy", then energy is also an "antiengergy", and that's not possible within THIS universe, because the entire universe would eventually (or, at least, less eventually than it already may happen) be destroyed by itself...in fact, I dare say we'd be in for one helluva light show! :)

Antienergy is probably energy from another universe that, from time to time, seeps into our own, but the laws of our universe (and perhaps all universes) are set up to prevent this from happening very often, or for long periods of time, to make sure all universes everywhere don't destroy each other in the blink of an eye...though whose blink are we timing? ;)