  We introduce a useful approach to find asymptotically explicit expressions
for the Casimir free energy at large temperature. The resulting expressions
contain the classical terms as well as the few first terms of the corresponding
heat-kernel expansion, as expected. This technique works well for many familiar
configurations in Euclidean as well as non-Euclidean spaces. By utilizing this
approach, we provide some new numerically considerable results for the Casimir
pressure in some rectangular ideal-metal cavities. For instance, we show that
at sufficiently large temperature, the Casimir pressure acting on the sidewalls
of a rectangular tube can be up to twice that of the two parallel planes. We
also apply this technique for calculating the Casimir free energy on a 3-torus
as well as a 3-sphere. We show that a nonzero mass term for both scalar and
spinor fields as well on the torus as on the sphere, violates the third law of
thermodynamics. We obtain some negative values for the Casimir entropy on the
3-torus as well as on the 3-sphere. We speculate that these negative Casimir
entropies can be interpreted thermodynamically as an instability of the vacuum
state at finite temperatures
