Thermal Casimir effect in closed Friedmann universe revisited

We reconsider Casimir free energy and internal energy at nonzero temperature in the static Einstein and closed Friedmann universe. It is shown that the Casimir free energy is given by the difference between the free energy of a topologically nontrivial manifold and a tangential Minkowski space-time. We derive exact expressions for the Casimir free energy, internal energy and pressure in Einstein and Friedmann universes in terms of single sums. The Casimir entropy is shown to satisfy the Nernst heat theorem. Exact expressions for corresponding total quantities in the Einstein universe are obtained from the Casimir ones by adding a contribution of the black-body radiation. The asymptotic expressions for the Casimir free energy and internal energy at both high and low temperature are shown to be in direct analogy with those for two ideal-metal plates and an ideal-metal spherical shell. Specifically, at high temperature the classical limit holds.