  The thermodynamic entropy of an isolated system is given by its von Neumann
entropy. Over the last few years, there is an intense activity to understand
thermodynamic entropy from the principles of quantum mechanics. More
specifically, is there a relation between the (von Neumann) entropy of
entanglement between a system and some (separate) environment is related to the
thermodynamic entropy? It is difficult to obtain the relation for many body
systems, hence, most of the work in the literature has focused on small number
systems. In this work, we consider black-holes --- that are simple yet
macroscopic systems --- and show that a direct connection could not be made
between the entropy of entanglement and the Hawking temperature. In this work,
within the adiabatic approximation, we explicitly show that the Hawking
temperature is indeed given by the rate of change of the entropy of
entanglement across a black hole's horizon with regard to the system energy.
This is yet another numerical evidence to understand the key features of black
hole thermodynamics from the viewpoint of quantum information theory.
