  The thermodynamics of Maxwell-Dilaton (dirty) black holes has been
extensively studied. It has served as a fertile ground to test ideas about
temperature through various definitions of surface gravity. In this paper, we
make an independent analysis of this black hole solution in both, Einstein and
Jordan, frames. We explore a set of definitions for the surface gravity and
observe the different predictions they make for the near extremal configuration
of this black hole. Finally, motivated by the singularity structure in the
interior of the event horizon, we use a holographic argument to remove the
micro-states from the disconnected region of this solution. In this manner, we
construct a frame independent entropy from which we obtain a temperature which
agrees with the standard results in the non-extremal regime, and has a
desirable behaviour around the extremal configurations according to the third
law of black hole mechanics.
