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We consider the finite temperature Casimir interaction between two Dirichletspheres in $(D+1)$-dimensional Minkowski spacetime. The Casimir interactionfree energy is derived from the zero temperature Casimir interaction energy viathe Matsubara formalism. In the high temperature region, the Casimirinteraction is dominated by the term with zero Matsubara frequency, and it isknown as the classical term since this term is independent of the Planckconstant $\hbar$. Explicit expression of the classical term is derived and itis computed exactly using appropriate similarity transforms of matrices. Wethen compute the small separation asymptotic expansion of this classical termup to the next-to-leading order term. For the remaining part of the finitetemperature Casimir interaction with nonzero Matsubara frequencies, we obtainits small separation asymptotic behavior by applying certain prescriptions tothe corresponding asymptotic expansion at zero temperature. This gives us aleading term that is shown to agree precisely with the proximity forceapproximation at any temperature. The next-to-leading order term at anytemperature is also derived and it is expressed as an infinite sum overintegrals. To obtain the asymptotic expansion at the low and medium temperatureregions, we apply the inverse Mellin transform techniques. In the lowtemperature region, we obtain results that agree with our previous work on thezero temperature Casimir interaction.

Finite temperature Casimir interaction between spheres in $(D+1)$-dimensional spacetime: Exact computations and asymptotic expansions