  A review of computations of free energy for Gibbs states on stationary but
not static gravitational and gauge backgrounds is given. On these backgrounds
wave equations for free fields are reduced to eigen-value problems which depend
non-linearly on the spectral parameter. We present a method to deal with such
problems. In particular, we demonstrate how some results of the spectral theory
of second order elliptic operators, such as heat kernel asymptotics, can be
extended to a class of non-linear spectral problems. The method is used to
trace down the relation between the canonical definition of the free energy
based on summation over the modes and the covariant definition given in
Euclidean quantum gravity. As an application, high-temperature asymptotics of
the free energy and of the thermal part of the stress-energy tensor in the
presence of rotation are derived. We also discuss statistical mechanics in the
presence of Killing horizons where canonical and Euclidean theories are related
in a non-trivial way.
