Physicist: Far be it for me to tell a god what it can do.
Without knowing more about the exact properties of gods, there’s not much to work with. The best way to answer this is to get a few thousand gods, and see if they last forever. If most or all of them do, then there’s a good chance that any particular god/being could last forever. This kind of eternity-based science requires some patience.
Unfortunately, the two tricks at a scientist’s disposal are observation and experiment. Even logic and inference have to step aside for experimental results (or more precisely, the assumptions that the logic is based on has to step aside). And there have been times when our “obvious” base assumptions, like “absolute time” (time is the same for everything) or the “geocentric hypothesis” (we’re totally not being flung around space at thousands of miles per hour), have been beaten down by overwhelming evidence. But gods (at least the god of the gaps) don’t seem to be observable.
I’m sure you’ve noticed that very few things seem to last forever (games of Risk, Iron Maiden, etc). Those things that we know of that have lasted, intact, the 13 or so billions years since the universe began tend to be very simple, like individual protons. As for things that existed “before the universe” (even assuming that makes any physical sense), nothing can be said at all. If it’s not in the universe, then it can’t be observed or experimented with and, scientifically speaking, that means we’re s.o.l.