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We derive rigorously explicit formulas of the Casimir free energy at finitetemperature for massless scalar field and electromagnetic field confined in aclosed rectangular cavity with different boundary conditions by zetaregularization method. We study both the low and high temperature expansions ofthe free energy. In each case, we write the free energy as a sum of apolynomial in temperature plus exponentially decay terms. We show that the freeenergy is always a decreasing function of temperature. In the cases of masslessscalar field with Dirichlet boundary condition and electromagnetic field, thezero temperature Casimir free energy might be positive. In each of these cases,there is a unique transition temperature (as a function of the side lengths ofthe cavity) where the Casimir energy change from positive to negative. When thespace dimension is equal to two and three, we show graphically the dependenceof this transition temperature on the side lengths of the cavity. Finally wealso show that we can obtain the results for a non-closed rectangular cavity byletting the size of some directions of a closed cavity going to infinity, andwe find that these results agree with the usual integration prescriptionadopted by other authors.

Finite temperature Casimir energy in closed rectangular cavities: a rigorous derivation based on zeta function technique