Higher dimensional rotating black hole solutions in quadratic $f(R)$ gravitational theory and the conserved quantities

Abstract: We explore the quadratic form of the $f(R)=R+bR^2$ gravitational theory to derive rotating $N$-dimensions black hole solutions with $a_i, i\geq 1$ rotation parameters. Here, $R$ is the Ricci scalar, and $b$ is the dimensional parameter. We assumed that the $N$-dimensional spacetime is static and has flat horizons with a zero curvature boundary. We investigated the physics of black holes by calculating the relations of physical quantities such as the horizon radius and mass. We also demonstrate that in the four-dimensional case,the higher-order curvature does not contribute to the black hole,i.e., black hole does not depend on the dimensional parameter $b$ whereas in the case of $N>4$, it depends on parameter $b$ owing to the contribution of the correction $R^2$ term. We analyze the conserved quantities, energy, and angular-momentum, of black hole solutions by applying the relocalization method. Additionally, we calculate the thermodynamic quantities such as temperature and entropy and examine the stability of black hole solutions locally and show that they have thermodynamic stability. Moreover, the calculations of entropy put a constraint on the parameter $b$ to be $b<\frac{1}{16\Lambda}$ to obtain a positive entropy.