We consider the toroidally compactified planar AdS-Schwarzschild solution to
4-dimensional gravity with negative cosmological constant. This has a flat
torus conformal boundary metric. We show that if the spatial part of the
boundary metric is deformed, keeping it static and the temperature and area
fixed, then assuming a static bulk solution exists, its energy is less than
that of the AdS-Schwarzschild solution. The proof is non-perturbative in the
metric deformation. While we expect the same holds for the free energy for
black hole solutions we are so far are not able to prove it. In the context of
AdS-CFT this implies a 3-dimensional holographic CFT on a flat spatial torus
whose bulk dual is AdS-Schwarzschild has a greater energy than if the spatial
geometry is deformed in any way that preserves temperature and area. This work
was inspired by previous results in free field theory, where scalars and
fermions in 3-dimensions have been shown to energetically disfavour flat space.