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Abstract We continue the effort of defining and evaluating the quantum entropy function for supersymmetric black holes in 4d N $$ \mathcal{N} $$ = 2 gauged supergravity, initiated in [1]. The emphasis here is on the missing steps in the previous localization analysis, mainly dealing with one-loop determinants for abelian vector multiplets and hypermultiplets on the non-compact space ℍ 2 × Σg with particular boundary conditions. We use several different techniques to arrive at consistent results, which have a most direct bearing on the logarithmic correction terms to the Bekenstein-Hawking entropy of said black holes.