Finite temperature Casimir effect of scalar field

We derive analytical expressions for the Helmholtz free energy, Casimir force, and Casimir entropy in one- and three-dimensional scalar fields subject to Dirichlet boundary conditions at finite temperature. The problem of negative Casimir entropy is examined in these systems, as well as for a scalar field confined within a three-dimensional (3D) spherical cavity. Our analysis shows that this apparent paradox arises under different regularization schemes involving distinct counterterms. We advocate against introducing any counterterms for the thermal corrections to the Casimir effect and predict that the thermal-fluctuation-induced Casimir force becomes repulsive in the high-temperature limit, specifically when aT/(πħv) > 0.2419 for the 3D scalar field.