  We study the spontaneous scalarization of spherically symmetric, static and
asymptotically Anti-de Sitter (aAdS) black holes in a scalar-tensor gravity
model with non-mininal coupling of the form $\phi^2\left(\alpha{\cal R} +
\gamma {\cal G}\right)$, where $\alpha$ and $\gamma$ are constants, while
${\cal R}$ and ${\cal G}$ are the Ricci scalar and Gauss-Bonnet term,
respectively. Since these terms act as an effective ``mass'' for the scalar
field, non-trivial values of the scalar field in the black hole space-time are
possible for {\it a priori} vanishing scalar field mass. In particular, we
demonstrate that the scalarization of an aAdS black hole requires the curvature
invariant $-\left(\alpha{\cal R} + \gamma {\cal G}\right)$ to drop below the
Breitenlohner-Freedman bound close to the black hole horizon, while it
asymptotes to a value well above the bound. The dimension of the dual operator
on the AdS boundary depends on the parameters $\alpha$ and $\gamma$ and we
demonstrate that -- for fixed operator dimension -- the expectation value of
this dual operator increases with decreasing temperature of the black hole,
i.e. of the dual field theory. When taking backreaction of the space-time into
account, we find that the scalarization of the black hole is the dual
description of a phase transition in a strongly coupled quantum system, i.e.
corresponds to a holographic phase transition. A possible application are
liquid-gas quantum phase transitions, e.g. in $^4$He. Finally, we demonstrate
that extremal black holes with $AdS_2\times S^2$ near-horizon geometry {\it
cannot support regular scalar fields on the horizon} in the scalar-tensor model
studied here.
