  We discuss spontaneous scalarization of the Reissner-Nordstr\"om black hole
in the presence of higher gauge field corrections that arise in the context of
string, as well as higher-dimensional more fundamental gravity theories. Our
theory admits the Reissner-Nordstr\"om solution at the scalar vacuum of the
theory ($\phi=0$) and we find that the higher order gauge field correction term
always result in the tachyonic instability of our system once the coupling
function satisfies the condition that its second derivative is positive at the
scalar vacuum. We find that the branches do not end with an extremal black
hole, rather with a singularity as indicated by the divergence of the
Kretschmann scalar. The black holes can be overcharged in the sense that they
may carry larger electric charge in comparison to their mass. Finally, these
solutions possess larger entropy at the event horizon radius when compared to
the Reissner-Nordstr\"om black hole, as well as to scalarized black holes
without the higher order gauge field terms, indicating in this way the
thermodynamic prefer-ability of our system, when compared to existing
literature, while they respect the energy conditions.

General Relativity and Quantum Cosmology;High Energy Physics - Theory