  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{2\Lambda r^2}
{3} $ and $h(r)=\frac{1}{2}-\frac{2M}{r}+\frac{q^2}{r^2}-\frac{2\Lambda r^2}
{3} $, respectively. The Ricci scalar of the (A)dS space-times has the form
${\textit R}=\frac{1+8r^2\Lambda}{r^2}$ and the form of ${\textit
F_1(R)}=\mp\frac{\textit 2\sqrt{R-8\Lambda}}{3M}$. 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.
