Uniqueness of non-trivial spherically symmetric black hole solution in special classes of F(R) gravitational theory

We show, in detail, that the only non-trivial black hole (BH) solutions for a neutral as well as a charged spherically symmetric space-times, using the class ${\textit F(R)}={\textit R}\pm{\textit F_1 (R)}$, must-have metric potentials in the form $h(r)=\frac{1}{2}-\frac{2M}{r}$ and $h(r)=\frac{1}{2}-\frac{2M}{r}+\frac{q^2}{r^2}$. These BHs have a non-trivial form of Ricci scalar, i.e., $R=\frac{1}{r^2}$ and the form of ${\textit F_1 (R)}=\mp\frac{\sqrt{\textit R}} {3M}$. We repeat the same procedure for (Anti-)de Sitter, (A)dS, space-time and got the metric potentials of neutral as well as charged in the form $h(r)=\frac{1}{2}-\frac{2M}{r}-\frac{2Λr^2} {3}$ and $h(r)=\frac{1}{2}-\frac{2M}{r}+\frac{q^2}{r^2}-\frac{2Λr^2} {3}$, respectively. The Ricci scalar of the (A)dS space-times has the form ${\textit R}=\frac{1+8r^2Λ}{r^2}$ and the form of ${\textit F_1(R)}=\mp\frac{\textit 2\sqrt{R-8Λ}}{3M}$. We calculate the thermodynamical quantities, Hawking temperature, entropy, quasi-local energy, and Gibbs-free energy for all the derived BHs, that behaves asymptotically as flat and (A)dS, and show that they give acceptable physical thermodynamical quantities consistent with the literature. Finally, we prove the validity of the first law of thermodynamics for those BHs.

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