Black hole thermodynamics in a time dependent metric For a time dependent space time metric, to get the thermodynamics, does the standard procedure of Wick rotating the time, and then calculating the free energy, work ?
 A: In principle it should not. The problem with the Wick rotation is that what you are doing is embedding the Lorentzian manifold in a complex manifold of which it is a slice, and then looking for a different slice with Riemannian signature. In general there is no such Riemmanian slice, and even if it does exist it need not be unique.
There is an old paper from Wald showing that for globally static spacetimes, such as Schwarzschild, everything goes smoothly, there is a unique slice and you can go on.
I'm not familiar with further improvements on this, besides this paper which gives necessary conditions for a spacetime to admit the Riemannian counterpart (very strong conditions indeed, the spacetime must posses totally geodesic three dimensional submanifold).
So you see that in general you have no reason to expect the Wick rotation to still work. But it may be the case that one can construct an example of time-evolving black hole that still has the Riemannian section. In any case it will propably not mantain the KMS condition (the periodicity in time of Green's functions) so that there is no well-defined temperature.
