  We study spontaneous scalarization of electrically charged extremal black
holes in $D\geq 4$ spacetime dimensions. Such a phenomenon is caused by the
symmetry breaking due to quartic interactions of the scalar -- Higgs potential
and Stueckelberg interaction with electromagnetic and gravitational fields,
characterized by the couplings $a$ and $b$, respectively. We use the entropy
representation of the states in the vicinity of the horizon, apply the inverse
attractor mechanism for the scalar field, and analyze analytically the
thermodynamic stability of the system using the laws of thermodynamics. As a
result, we obtain that the scalar field condensates on the horizon only in
spacetimes which are asymptotically non-flat, $\Lambda \neq 0$ (dS or AdS), and
whose extremal black holes have non-planar horizons $k=\pm 1$, provided that
the mass $m$ of the scalar field belongs to a mass interval (area code)
different for each set of the boundary conditions specified by $(\Lambda ,k)$.
A process of scalarization describes a second order phase transition of the
black hole, from the extremal Reissner-Nordstr\"{o}m (A)dS one, to the
corresponding extremal hairy one. Furthermore, for the transition to happen,
the interaction has to be strong enough, and all physical quantities on the
horizon depend at most on the effective Higgs-Stueckelberg interaction
$am^2-2b$. Most of our results are general, valid for any parameter and any
spacetime dimension.
