In this paper, we study thermostatistical properties of a photon gas in the framework of two deformed special relativity models defined by the cosmological coordinatizations of the de Sitter (dS) and anti-de Sitter (AdS) momentum spaces. The dS model is a doubly special relativity theory in which an ultraviolet length scale is invariant under the deformed Lorentz transformations. For the case of the AdS model, however, the Lorentz symmetry breaks at the high energy regime. We show that the existence of a maximal momentum in dS momentum space leads to maximal pressure and temperature at the thermodynamical level, while maximal internal energy and entropy arise for the case of the AdS momentum space due to the existence of a maximal kinematical energy. These results show that the thermodynamical duality of these models is very similar to their well-known kinematical duality.
