Exploring the Impact of $\Lambda = a \cdot e^\pi$ as a Cosmological Constant on Black Hole Solutions in General Relativity and $f(R)$ Gravity

In this paper, we investigate the cosmological constant $\Lambda = a \cdot e^\pi$, where $a$ is an algebraic parameter, and demonstrate its role in $f(R)$ gravity as an indicator of the transcendental form. We analyze the Schwarzschild and Kerr-Newman black holes under this cosmological constant and show that they satisfy the $\mathrm{SO} \times \mathrm{R}$ symmetry. Additionally, we prove that $f(R)$ gravity with this transcendental form also adheres to $\mathrm{SO} \times \mathrm{R}$ symmetry.