  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,
differently to the conformal limit, fulfil Bekenstein's and Verlinde's bounds
in the physical region.
