  A deformed relativistic kinematics can be understood within a geometrical
framework through a maximally symmetric momentum space. However, when
considering this kind of approach, usually one works in a flat spacetime and in
a curved momentum space. In this paper, we will discuss a possible
generalization to take into account both curvatures and some possible
observable effects. We will first explain how to construct a metric in the
cotangent bundle in order to have a curved spacetime with a nontrivial geometry
in momentum space and the relationship with an action in phase space
characterized by a deformed Casimir. Then, we will study within this proposal
two different space-time geometries. In the Friedmann-Robertson-Walker
universe, we will see the modifications in the geodesics (redshift, luminosity
distance and geodesic expansion) due to a momentum dependence of the metric in
the cotangent bundle. Also, we will see that when the spacetime considered is a
Schwarzschild black hole, one still has a common horizon for particles with
different energies, differently from a Lorentz invariance violation case.
However, the surface gravity computed as the peeling off of null geodesics is
energy dependent.
