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We investigate Casimir effect as well as thermal Casimir effect for a pair of parallel perfectly plates placed in general stationary space–time background. It is found that the Casimir energy is influenced by the 00-component of metric and the corresponding quantity in dragging frame. We give a scheme to renormalize thermal correction to free energy in curved space–time. It is shown that the thermal corrections to Casimir thermodynamic quantities not only depend on the proper temperature and proper geometrical parameters of the plates, but also on the determinant of space–time metric. Keywords: Casimir effect, Thermal Casimir effect, Stationary space–time