  It is known that general relativity (GR) theory is not consistent with the
latest observations. The modified gravity of GR known as $\mathrm{f(R)}$ where
$\mathrm{R}$ is the Ricci scalar, is considered to be a good candidate for
dealing with the anomalies present in classical GR. In this context, we study
static rotating uncharged anti-de Sitter and de Sitter (AdS and dS) black holes
(BHs) using $\mathrm{f(R)}$ theory without assuming any constraints on the
Ricci scalar or on $\mathrm{f(R)}$. We derive BH solutions depend on the
convolution function and deviate from the AdS/dS Schwarzschild BH solution of
GR. Although the field equations have no dependence on the cosmological
constant, the BHs are characterized by an effective cosmological constant that
depends on the convolution function. The asymptotic form of this BH solution
depends on the gravitational mass of the system and on extra terms that lead to
BHs being different from GR BHs but to correspond to GR BHs under certain
conditions. We also investigate how these extra terms are responsible for
making the singularities of the invariants milder than those of the GR BHs. We
study some physical properties of the BHs from the point of view of
thermodynamics and show that there is an outer event horizon in addition to the
inner Cauchy horizons. Among other things, we show that our BH solutions
satisfy the first law of thermodynamics. To check the stability of these BHs we
use the geodesic deviations and derive the stability conditions. Finally, using
the odd-type mode it is shown that all the derived BHs are stable and have a
radial speed equal to one.
