Black hole entropy and moduli-dependent species scale

We provide a moduli-dependent definition of species scale in quantum gravity based on black hole arguments. Concretely, it is derived from of a lower bound on the entropy of extremal black holes with higher curvature corrections, which ensures that the black hole can be reliably described within the effective theory. By demanding that our definition coincides with a recent proposal for a moduli-dependent species scale motivated from the topological string, we conclude that the conjecture ${\mathcal{Z}}_{BH}=\mathrm{\mid }{\mathcal{Z}}_{\mathrm{t}\mathrm{o}\mathrm{p}}{\mathrm{\mid }}^2$ relating the black hole to the topological string partition functions should hold, at least within the regime of validity of our analysis.