  We compute the area term contribution to black holes' entanglement entropy
(using the conical technique) for a class of local or weakly non-local
super-renormalizable gravitational theories coupled to matter. For the first
time, we explicitly prove that all the beta functions in the proposed theory,
except for the cosmological constant, are identically zero in cut-off
regularization scheme and not only in dimensional regularization scheme. In
particular, we show that there is no divergence quadratic in cut-off and hence
there is no contribution to the beta function of the Newton constant. As a
consequence of this result, we argue that in these theories of gravity conical
entropy is a sensible definition of physical entropy, in particular, it is
positive-definite and gauge-independent. On top of this the conical entropy,
being expressed only in terms of the classical Newton constant, turns out to be
finite and naturally coincides with Bekenstein-Hawking entropy. Finally, we
propose a theory in which the renormalization of the Newton constant is
entirely due to the Standard Model matter, arguing that such a contribution
does not give the usual interpretational problems of conical entropy discussed
in the literature.
