  In a quantum gravity theory, spacetime at mesoscopic scales can acquire a
novel structure very different from the classical concept of general
relativity. A way to effectively characterize the quantum nature of spacetime
is through a momentum dependent space-time metric. There is a vast literature
showing that this geometry is related to deformed relativistic kinematics,
which is precisely a way to capture residual effects of a quantum gravity
theory. In this work, we study the notion of surface gravity in a momentum
dependent Schwarzschild black hole geometry. We show that using the two main
notions of surface gravity in general relativity we obtain a momentum
independent result. However, there are several definitions of surface gravity,
all of them equivalent in general relativity when there is a Killing horizon.
We show that in our scheme, despite the persistence of a Killing horizon, these
alternative notions only agree in a very particular momentum basis, obtained in
a previous work, so further supporting its physical relevance.
