  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.
