  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 simple Smarr-type relations involving
the boundary stress tensor..
