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