  Exact calculations are given for the Casimir energy for various fields in
$R\times S^3$ geometry. The Green's function method naturally gives a result in
a form convenient in the high-temperature limit, while the statistical
mechanical approach gives a form appropriate for low temperatures. The
equivalence of these two representations is demonstrated. Some discrepancies
with previous work are noted. In no case, even for ${\cal N}=4$ SUSY, is the
ratio of entropy to energy found to be bounded.
