  Taking the Noether gauge symmetry approach into account, we find spherically
symmetric static black hole solutions of the non-minimal gauge-gravity
Lagrangian of the $\mathcal{R}^\beta F^2$ model. At first, we consider a system
of differential equations for the general non-minimal couplings of
$Y(\mathcal{R})F^2$ type, and then, we regard a particular $\mathcal{R}^\beta
F^2$ non-minimal model to find the exact black hole solution and analyze its
symmetries. As the next step, we calculate the thermodynamical quantities of
the black hole and study its interesting behavior. Besides, we address thermal
stability and examine the possibility of the van der Waals-like phase
transition.
