Noether gauge symmetry approach applying for the non-minimally coupled gravity to the Maxwell field

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.