  We numerically study the topologically twisted index of several
three-dimensional supersymmetric field theories on a genus $g$ Riemann surface
times a circle, $\Sigma_g\times S^1$. We show that for a large class of
theories with leading term of the order $N^{3/2}$, where $N$ is generically the
rank of the gauge group, there is a universal logarithmic correction of the
form $\frac{g-1}{2} \log N$. We explain how this logarithmic subleading
correction can be obtained as a one-loop effect on the dual supergravity theory
for magnetically charged, asymptotically AdS$_4\times M^7$ black holes for a
large class of Sasaki-Einstein manifolds, $M^7$. The matching of the
logarithmic correction relies on a generic cohomological property of $M^7$ and
it is independent of the black hole charges. We argue that our supergravity
results apply also to rotating, electrically charged asymptotically
AdS$_4\times M^7$ black holes. We present explicitly the quiver gauge theories
and the gravity side corresponding to $M^7=N^{0,1,0}, V^{5,2}$ and $Q^{1,1,1}$.
