Thermal Casimir Effect in the Einstein Universe with a Spherical Boundary

In the present paper, we investigate thermal fluctuation corrections to the vacuum energy at zero temperature of a conformally coupled massless scalar field, whose modes propagate in the Einstein universe with a spherical boundary, characterized by both Dirichlet and Neumann boundary conditions. Thus, we generalize the results found in the literature in this scenario, which has considered only the vacuum energy at zero temperature. To do this, we use the generalized zeta function method plus Abel-Plana formula and calculate the renormalized Casimir free energy as well as other thermodynamics quantities, namely, internal energy and entropy. For each one of them, we also investigate the limits of high and low temperatures. At high temperatures, we found that the renormalized Casimir free energy presents classical contributions, along with a logarithmic term. Also in this limit, the internal energy presents a classical contribution and the entropy a logarithmic term, in addition to a classical contribution as well. Conversely, at low temperatures, it is demonstrated that both the renormalized Casimir free energy and internal energy are dominated by the vacuum energy at zero temperature. It is also demonstrated that the entropy obeys the third law of thermodynamics