Vacuum confinement at finite temperature for scalar QED in magnetic field and deformed boundary condition

We investigate the Casimir effect at finite temperature for a charged scalar field in the presence of an external uniform and constant magnetic field, perpendicular to the Casimir plates. We have used a boundary condition characterized by a deformation parameter $\theta$; for $\theta=0$ we have a periodic condition and for $\theta=\pi$, an antiperiodic one, for intermediate values, we have a deformation. The temperature was introduced using the imaginary time formalism and both the lagrangian and free energy were obtained from Schwinger proper time method for computing the effective action. We also computed the permeability and its asymptotic expressions for low and high temperatures.