by Hamiltonian formalism [EntropyLov1]{} (for its generalization for arbitrarily high order derivatives of the curvature, see [@EntropyLov2]), yielding $$S=\frac{{{V_{n-1}}}}{4}r_{+}^{n-1}\left( {1+\frac{{2\left( {n-1}\right) k}}{{%
\left( {n-3}\right) }}\frac{\alpha }{{r_{+}^{2}}}+\frac{{\left( {n-1}\right)
{k^{2}}}}{{\left( {n-5}\right) }}\frac{{{\alpha ^{2}}}}{{r_{+}^{4}}}}\right)
,  \label{Entropy}$$It is clear that Eq. (\[Entropy\]) reproduces the area law for Einstein gravity ($\alpha \longrightarrow 0$).

In order to obtain the electric charge, we calculate the flux of the electromagnetic field at infinity. It is easy to show that $$Q=\frac{V_{n-1}}{4\pi }q.  \label{Charge}$$Eq. (\[Charge\]) confirms that the electric charge does not depend on the nonlinearity parameter.

In order to calculate the electric potential of the black holes, one should consider a reference. Considering nonzero component of the gauge potential (or the electric field), one finds that for $r\longrightarrow \infty $, both the gauge potential and the electric field vanishes. Therefore, it is natural to calculate the electric potential of the event horizon of black holes, $r_{+}$, with respect to the infinity as reference [@Potential]. We obtain $$\begin{aligned}
\Phi &=&A_{\mu }\chi ^{\mu }\left\vert _{r\rightarrow \infty }-A_{\mu }\chi
^{\mu }\right\vert _{r=r_{+}}=  \nonumber \\
&&\left\{
\begin{array}{cc}
\frac{{\beta {r_{+}}\sqrt{L{w_{+}}}}}{{2(n-2)(3n-4)}}\left[ {(n-1)\,{\zeta
_{+}L}_{W+}+3n-4}\right] , & \text{ENED} \\
-\frac{{2{\beta ^{2}}{r_{+}}^{n}}}{{nq}}\left( {{\eta _{+}}-1}\right) , &
\text{LNED}%
\end{array}%
\right. .  \label{Pot}\end{aligned}$$

$$\zeta _{+}={}_{2}{F_{1}}\left( {[1],\,\left[ {\frac{{5n-6}}{{2(n-1)}}}\right]
,\,}\frac{{L}_{W+}}{{2(n-1)}}\right) ,$$

$$\eta 