  The first quantum corrections to the free energy for massive fields in
$D$-dimensional space-times of the form $\R\times\R^+\times\M^{N-1}$, where
$D=N+1$ and $\M^{N-1}$ is a constant curvature manifold, is investigated by
means of the $\zeta$-function regularization. It is suggested that the nature
of the divergences, which are present in the thermodynamical quantities, might
be better understood making use of the conformal related optical metric and
associated techniques. The general form of the horizon divercences of the free
energy is obtained as a function of free energy densities of fields having
negative square masses (absence of the gap in the Laplace operator spectrum) on
ultrastatic manifolds with hyperbolic spatial section $H^{N-2n}$ and of the
Seeley-DeWitt coefficients of the Laplace operator on the manifold $\M^{N-1}$.
Furthermore, recurrence relations are found relating higher and lower
dimensions. The cases of Rindler space, where $\M^{N-1}=\R^{N-1}$ and very
massive $D$-dimensional black holes, where $\M^{N-1}=S^{N-1}$ are treated as
examples. The renormalization of the internal energy is also discussed.

High Energy Physics - Theory;General Relativity and Quantum Cosmology