Quadratic and cubic spherically symmetric black holes in the modified teleparallel equivalent of general relativity: energy and thermodynamics

In \cite{Bahamonde:2019zea}, a spherically symmetric black hole (BH) was derived from the quadratic form of $f(T)$. Here we derive associated energy, invariants curvature, and torsion this BH demonstrate that higher-order contribution renders singularity weaker compared with Schwarzschild general relativity (GR). Moreover, calculate thermodynamic quantities reveal effect higher--order on these quantities. Therefore, new cubic $f(T)=T+\epsilon\Big[\frac{1}{2}\alpha T^2+\frac{1}{3}\beta T^3\Big]$, where $\epsilon<<1$, $\alpha$, $\beta$ are constants. The is characterized by two constants $\alpha$ in addition to $\epsilon$. At $\epsilon=0$ return GR. We study physics solutions via same procedure applied for BH. torsion, $\frac{1}{2}\alpha T^3$, may afford an interesting physics.