Lib, if I recall my thermo correctly, the energy available to do work isn’t the Gibbs free energy, it’s the Helmholtz free energy. This is given by A = U - TS = H - PV - TS = G - PV.
Also if I recall my thermo correctly, the equation you have as written is only true for constant temperatures, and the universe doesn’t have constant temperature, of course.
All that aside, you ask an excellent question, and I have no idea what the answer is. It seems intuitively obvious that a Big Crunch would result, if nothing else, in a decrease in the total entropy of the universe, which is strictly verboten, so either I’m failing to appreciate a fundamental point about what the entropy of the universe is, or thermodynamic laws don’t work in extreme situations like a Big Bang/Big Crunch, or both. And I’m not really sure what the answer is.
It seems like what has to happen if the Big Crunch picture is correct is that the Helmholtz energy has to be reset. How that happens, I don’t know. But a key point is that the Helmholtz energy tells you how much energy is available to do external work, if I remember correctly (Atkins is a little vague on this point, and I can’t find my notes). If it’s the amount of energy for external work, then there’s really no problem, as of course the universe by definition can’t be involved in doing external work.
I realize this is profoundly unhelpful, but it’s the best I can do right now. Sorry.