  The classical and continuum limit of a quantum gravitational setting could
lead, at mesoscopic regimes, to a very different notion of geometry w.r.t. the
pseudo-Riemannian one of special and general relativity. A possible way to
characterize this modified space-time 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 space-time coordinates, compatible with the relativistic
deformed kinematics of doubly special relativity.
