  We study the Casimir pressure between two similar plates of finite thickness
kept at different temperatures in the case when the dielectric permittivity of
the plates depends on temperature. It is suggested to consider the dielectric
permittivity at two different temperatures as the permittivities of two
dissimilar bodies, thus allowing to apply the theory of Casimir forces out of
thermal equilibrium developed earlier in the literature. Following this
approach, we show that, in addition to the equilibrium contribution to the
nonequilibrium Casimir pressure, a proper nonequilibrium contribution arises
for temperature-dependent dielectric permittivities. Furthermore, the
equilibrium contribution in this case does not equal the mean of the
equilibrium Casimir pressures at the temperatures of the plates. As an
application, the total nonequilibrium Casimir pressure between two gold plates
and between two titanium plates is calculated as a function of the plate
thickness and their separation, using the Drude and the plasma model. For plate
separations ranging from 0.5 to 2 micrometers, the relative difference between
the theoretical predictions for these two models reaches 39%. The proper
nonequilibrium term may be as large as 4% of the magnitude of the total
nonequilibrium pressure.
