  A possible way to capture the effects of quantum gravity in spacetime at a
mesoscopic scale, for relatively low energies, is through an energy dependent
metric, such that particles with different energies probe different spacetimes.
In this context, a clear connection between a geometrical approach and
modifications of the special relativistic kinematics has been shown in the last
few years. In this work, we focus on the geometrical interpretation of the
relativistic deformed kinematics present in the framework of doubly special
relativity, where a relativity principle is present. In this setting, we study
the effects of a momentum dependence of the metric for a uniformly accelerated
observer. We show how the local Rindler wedge description gets affected by the
proposed observer dependent metric, while the local Rindler causal structure is
not, leading to a standard local causal horizon thermodynamic description. For
the proposed modified metric, we can reproduce the derivation of Einstein's
equations as the equations of state for the thermal Rindler wedge. The
conservation of the Einstein tensor leads to the same privileged momentum basis
obtained in other works of some of the present authors, so supporting its
relevance.
