  We study the spectrum of scalar $l=0$ quasi-normal frequencies of
Anti-de-Sitter hairy black holes in four and five dimensional designer gravity
theories of the Einstein-scalar type, arising as consistent truncations of $
\mathcal{N} = 8$ gauged supergravity. In the dual field theory, such hairy
black holes represent thermal states in which the operator corresponding to the
bulk scalar field is condensed, due to the multi-trace deformation associated
with non-standard boundary conditions. We show that, in a particular class of
models, the effective potential describing the vacua of the deformed dual
theory can be identified, at large values of the condensate, with the
deformation plus the conformal coupling of the condensate to the curvature of
the boundary geometry. In this limit, we show that the least damped
quasi-normal frequency of the corresponding hairy black holes can be accurately
predicted by the curvature of the effective potential describing the field
theory at finite entropy.
