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Taking the Noether gauge symmetry approach into account, spherically symmetric static black hole solutions of the non‐minimal gauge‐gravity Lagrangian of theR  β    F  2    $\mathcal {R}^\beta F^2$    model is found. At first, a system of differential equations for the general non‐minimal couplings ofY   (  R  )    F  2    $Y(\mathcal {R})F^2$    type is considered, and then, it is regarded as a particularR  β    F  2    $\mathcal {R}^\beta F^2$    non‐minimal model to find the exact black hole solution and analyze its symmetries. As the next step, the thermodynamical quantities of the black hole and its interesting behavior is calculated. Besides, thermal stability and examining the possibility of the van der Waals‐like phase transition is addressed.

Noether Gauge Symmetry Approach Applying for the Non‐Minimally Coupled Gravity to the Maxwell Field