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It is well known that General Relativity cannot be considered under thestandard of a perturbatively renormalizable quantum field theory, butasymptotic safety is taken into account as a possibility for the formulation ofgravity as a non-perturbative renormalizable theory. Recently, the entropyargument has however stepped into the discussion claiming for a "no-go" to theasymptotic safety argument. In this paper, we present simple counter-examples,considering alternative theories of gravity, to the entropy argument as furtherindications, among others, on the possible flows in the assumptions on whichthe latter is based on. We consider different theories, namely curvature basedextensions of General Relativity as $f(R)$, $f(\mathcal{G})$, extensions ofTeleparallel Gravity as $f(T)$, and Horava-Lifshitz gravity, working out theexplicit spherically symmetric solutions in order to make a comparison betweenpower counting and the entropy argument. Even in these cases, inconsistenciesare found.

Renormalizability of alternative theories of gravity: differences between power counting and entropy argument