Universality of small black hole instability in AdS/CFT

$AdS_5$ type IIb supergravity compactifications on five-dimensional Einstein manifolds ${\cal V}_5$ realize holographic duals to four-dimensional conformal field theories. Black holes in such geometries are dual to thermal states in these CFTs. When black holes become sufficiently small in (global) $AdS_5$, they are expected to suffer Gregory-Laflamme instability with respect to localization on ${\cal V}_5$. Previously, the instability was demonstrated for gravitational dual of ${\cal N}=4$ SYM, where ${\cal V}_5=S^5$. We extend stability analysis to arbitrary ${\cal V}_5$. We point out that the quasinormal mode equation governing the instabilities is universal. The precise onset of the instability is ${\cal V}_5$-sensitive, as it is governed by the lowest non-vanishing eigenvalue $\lambda_{min}$ of its Laplacian.