  We consider (2+1)-QFT at finite temperature on a product of time with a
static spatial geometry. The suitably defined difference of thermal vacuum free
energy for the QFT on a deformation of flat space from its value on flat space
is a UV finite quantity, and for reasonable fall-off conditions on the
deformation is IR finite too. For perturbations of flat space we show this free
energy difference goes quadratically with perturbation amplitude and may be
computed from the linear response of the stress tensor. As an illustration we
compute it for a holographic CFT finding that at any temperature, and for any
perturbation, the free energy decreases. Similar behaviour was previously found
for free scalars and fermions, and for unitary CFTs at zero temperature,
suggesting (2+1)-QFT may generally energetically favour a crumpled spatial
geometry. We also treat the deformation in a hydrostatic small curvature
expansion relative to the thermal scale. Then the free energy variation is
determined by a curvature correction to the stress tensor and for these
theories is negative for small curvature deformations of flat space.
