  We derive rigorously explicit formulas of the Casimir free energy at finite
temperature for massless scalar field and electromagnetic field confined in a
closed rectangular cavity with different boundary conditions by zeta
regularization method. We study both the low and high temperature expansions of
the free energy. In each case, we write the free energy as a sum of a
polynomial in temperature plus exponentially decay terms. We show that the free
energy is always a decreasing function of temperature. In the cases of massless
scalar field with Dirichlet boundary condition and electromagnetic field, the
zero 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 of
the cavity) where the Casimir energy change from positive to negative. When the
space dimension is equal to two and three, we show graphically the dependence
of this transition temperature on the side lengths of the cavity. Finally we
also show that we can obtain the results for a non-closed rectangular cavity by
letting the size of some directions of a closed cavity going to infinity, and
we find that these results agree with the usual integration prescription
adopted by other authors.
