Moduli-dependent one-loop entropy of hyperbolic BPS black hole in AdS$_4$

We study one-loop logarithmic corrections to the entropy of static hyperbolic BPS black holes in asymptotically AdS$_4$ spacetime. Our analysis is carried out in a consistent real-scalar truncation of ${\cal N}=2$ Fayet-Iliopoulos gauged supergravity specified by the prepotential $F=-i X^0 X^1$, which corresponds to an Einstein-Dilaton-Maxwell theory with a nontrivial scalar potential. In this model, the classical BPS attractor mechanism exhibits flat directions, leaving scalar moduli on the black hole horizon unfixed, while the Bekenstein-Hawking entropy depends only on the charges. We show that the resulting one-loop correction to the black hole entropy acquires a nontrivial dependence on the horizon moduli and induces an effective quantum potential that dynamically stabilizes them at a preferred value. Our results provide an explicit and concrete realization of quantum lifting of classical attractor flat directions in gauged supergravity.