Thermal corrections to the Casimir energy in a general weak gravitational field

We calculate finite temperature corrections to the energy of the Casimir effect of a two conducting parallel plates in a general weak gravitational field. After solving the Klein-Gordon equation inside the apparatus, mode frequencies inside the apparatus are obtained in terms of the parameters of the weak background. Using Matsubara's approach to quantum statistical mechanics gravity-induced thermal corrections of the energy density are obtained. Well-known weak static and stationary gravitational fields are analyzed and it is found that in the low temperature limit the energy of the system increases compared to that in the zero temperature case.