Entropy bounds for massive scalar field in positive curvature space

We consider a massive scalar field with arbitrary coupling in ${\mathbf{S}}^1\times {\mathbf{S}}^3$ space, which mimics the thermal expanding Universe, and calculate explicitly all relevant thermodynamical functions in the low- and high-temperature regimes, extending previous analysis of entropy bounds and entropy/energy ratios performed in the conformal case. For high temperatures, new mass-dependent entropy ratios are established which, different from the conformal limit, satisfy Bekenstein’s and Verlinde’s bounds in the physical region.