  The study of black hole physics revealed a fundamental connection between
thermodynamics, quantum mechanics, and gravity. Today, it is known that black
holes are thermodynamical objects with well-defined temperature and entropy.
Although black hole radiance gives us the mechanism from which we can associate
a well-defined temperature to the black hole, the origin of its entropy remains
a mystery. Here we investigate how the quantum fluctuations from the fields
that render the black hole its temperature contribute to its entropy. By using
the exact renormalization group equation for a self-interacting real scalar
field in a spacetime possessing a bifurcate Killing horizon, we find the
renormalization group flow of the total gravitational entropy. We show that
throughout the flow one can split the quantum field contribution to the entropy
into a part coming from the entanglement between field degrees of freedom
inside and outside the horizon and a part due to the quantum corrections to the
Wald entropy coming from the Noether charge. The renormalized black hole
entropy is shown to be constant throughout the flow while the balance between
the effective black hole entropy at low energies and the infra-red entanglement
entropy changes. A similar conclusion is valid for the Wald entropy part of the
total entropy. Additionally, our calculations show that there is no mismatch
between the renormalization of the coupling constants coming from the effective
action or the gravitational entropy, solving an apparent "puzzle" that appeared
to exist for interacting fields.
