  We study the finite temperature Casimir interaction between two concentric
cylinders. When the separation between the cylinders is much smaller than the
radii of the cylinders, the asymptotic expansions of the Casimir interaction
are derived. Both the low temperature and the high temperature regions are
considered. The leading terms are found to agree with the proximity force
approximations. The low temperature leading term of the temperature correction
is also computed and it is found to be independent of the boundary conditions
imposed on the larger cylinder.
