Exploring black hole mechanics in cotangent bundle geometries

The classical and continuum limit of a quantum gravitational setting could lead, at mesoscopic regimes, to a very different notion of geometry with respect to the pseudo-Riemannian one of special and general relativity. A possible way to characterize this modified spacetime notion is by a momentum-dependent metric, in such a way that particles with different energies could probe different spacetimes. Indeed, doubly special relativity theories, deforming the special relativistic kinematics while maintaining a relativity principle, have been understood within a geometrical context, by considering a curved momentum space. The extension of these momentum spaces to curved spacetimes and its possible phenomenological implications have been recently investigated. Following this line of research, we address here the first two laws of black holes thermodynamics in the context of a cotangent bundle metric, depending on both momentum and spacetime coordinates, compatible with the relativistic deformed kinematics of doubly special relativity.