Free energy and entropy in Rindler and de Sitter space-times

We investigate the free energy and entropy of Gaussian massive scalar field theory in static de Sitter space-time for arbitrary temperature. For inverse temperatures form $\ensuremath{\beta}=2\ensuremath{\pi}{2}^{k}$, $k\ensuremath{\in}\mathbf{Z}$, curvature units, we find explicit its derivatives with respect to There are two types contributions energy: one is ``area type'' can be attributed horizon, while other ``volume associated interior space-time. The latter contribution odd-dimensional case limit week (large mass or small Hubble constant) significantly depends on Namely, $\ensuremath{\beta}<2\ensuremath{\pi}$, behaves as ${F}_{\ensuremath{\beta}}^{\text{bulk}}\ensuremath{\sim}{e}^{\ensuremath{-}\ensuremath{\beta}m}$, $\ensuremath{\beta}>2\ensuremath{\pi}$ it ${F}_{\ensuremath{\beta}}^{\text{bulk}}\ensuremath{\sim}{e}^{\ensuremath{-}2\ensuremath{\pi}m}$. also show that even leading UV depend state theory, which very unusual. explain origin physical meaning these observations. As model example consider situation Rindler wedge flat