On the thermodynamics of periodic AdS black branes

We consider asymptotically AdS black brane solutions that are dual to CFTs with periodic dependence on the spatial directions, arising from either a spontaneous or an explicit breaking of translational symmetry. We derive a simple expression for the variation of the free-energy with respect to changing the periods. This explains some observations, based on numerics, that have arisen in the explicit construction of thermodynamically preferred black holes in the case that the spatial directions are infinite in extent and the symmetry is spontaneously broken. It also leads to Smarr-type relations involving the boundary stress tensor.