2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/149174We show that the partition function for a scalar field in a static spacetime background can be expressed as a functional integral in the corresponding optical space, and point out that the difference between this and the functional integral in the original metric is a Liouville type action. A general formula for the free energy is derived in the high temperature approximation and applied to various cases. In particular we find that thermodynamics in the extremal Reissner-Nordström space has extra singularities that make it ill-defined.10 pages, phyzzx (added references)High Energy Physics - TheoryGeneral Relativity and Quantum Cosmologytext