Spontaneous scalarization in (A)dS gravity at zero temperature

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