  Using recent results in four-derivative 5d $\mathcal{N}=2$ minimal gauged
supergravity, we evaluate the regularized on-shell action of the Euclidean
solution in this theory that admits a Lorentzian continuation to an AdS$_5$
black hole with one electric charge and two angular momenta. We focus on the
supersymmetric limit of this solution and employ holography to show that the
result can be expressed purely in terms of angular momentum fugacities and the
't Hooft anomalies for the $\mathrm{U}(1)_{\mathrm R}$ R-symmetry of the dual
4d $\mathcal{N}=1$ SCFT. This holographic calculation is in perfect agreement
with recent studies of the 4d $\mathcal{N}=1$ superconformal index ``on the
second sheet''. We illustrate the utility of these results in two classes of 4d
$\mathcal{N}=1$ holographic SCFTs that have 't Hooft anomalies with suitable
large $N$ behavior that leads to non-trivial corrections at first subleading
order in the $1/N$ expansion. We also explicitly calculate the Wald entropy of
the black hole solution and delineate the leading four-derivative corrections
to the Bekenstein-Hawking entropy.
