  We consider the finite temperature Casimir interaction between two Dirichlet
spheres in $(D+1)$-dimensional Minkowski spacetime. The Casimir interaction
free energy is derived from the zero temperature Casimir interaction energy via
the Matsubara formalism. In the high temperature region, the Casimir
interaction is dominated by the term with zero Matsubara frequency, and it is
known as the classical term since this term is independent of the Planck
constant $\hbar$. Explicit expression of the classical term is derived and it
is computed exactly using appropriate similarity transforms of matrices. We
then compute the small separation asymptotic expansion of this classical term
up to the next-to-leading order term. For the remaining part of the finite
temperature Casimir interaction with nonzero Matsubara frequencies, we obtain
its small separation asymptotic behavior by applying certain prescriptions to
the corresponding asymptotic expansion at zero temperature. This gives us a
leading term that is shown to agree precisely with the proximity force
approximation at any temperature. The next-to-leading order term at any
temperature is also derived and it is expressed as an infinite sum over
integrals. To obtain the asymptotic expansion at the low and medium temperature
regions, we apply the inverse Mellin transform techniques. In the low
temperature region, we obtain results that agree with our previous work on the
zero temperature Casimir interaction.
