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.

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