Casimir effect at finite temperature

The high-temperature expansion of the grand thermodynamic potential of a nonconformally invariant spin-0 gas in an arbitrary ultrastatic spacetime with boundary is given in terms of the Minakshisundaram-Pleijel coefficients of the heat-kernel and the zeta function of the spatial section. The general formula is then used to find the expansion in the case of a massive bosonic field subject to Dirichlet boundary conditions on hypercuboids in a flat n-dimensional spacetime. A detailed analysis of inhomogeneous multidimensional Epstein zeta functions is necessary and some new properties of them are derived. Finally the thermodynamics of the system is considered.