Stefan-Boltzmann law and Casimir effect for dark photons

In this paper, the existence of a massive dark photon, associated with a new gauge group is considered. The dark photon can be kinetically mixed with the photon. To study some applications, the thermo field dynamics formalism is used. Exploring the topological structure of this approach, the influence of dark photons on the Stefan-Boltzmann law and the Casimir effect at zero and finite temperature is calculated.