refer the reader to Ref. [@HendiJHEP; @HendiAnn]). Drawing the Carter-Penrose diagrams shows that the causal structure of the solutions are asymptotically well behaved.

Thermodynamic properties and thermal stability [Thermodynamics]{}
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In this section, we calculate the conserved and thermodynamic quantities, and check the first law of black hole thermodynamics. Then we perform the stability criterion.

At first, we apply the definition of surface gravity to obtain the Hawking temperature $$T=\frac{1}{2\pi }\sqrt{-\frac{1}{2}\left( \nabla _{\mu }\chi _{\nu }\right)
\left( \nabla ^{\mu }\chi ^{\nu }\right) },  \label{T1}$$where $\chi $ is the temporal Killing vector, $\partial _{t}$. One obtains$$T=\frac{f^{\prime }(r_{+})}{4\pi }=\frac{(n-1)k\left[ 3{\left( {n-2}\right)
r_{+}^{4}+3\left( {n-4}\right) {k}\alpha {r_{+}^{2}}+\left( {n-6}\right) {%
\alpha ^{2}}}\right] -6{\Lambda }r_{+}^{6}+3{{\beta ^{2}}r_{+}^{6}}\Psi }{{%
12\pi (n-1){r_{+}{\left( {r_{+}^{2}+k\alpha }\right) }^{2}}}},  \label{T2}$$where$$\Psi =\left\{
\begin{array}{cc}
\frac{{2\left( {1-}L_{W+}\right) }}{\sqrt{L_{W+}}}\sqrt{\Gamma _{+}^{2}-1}-1,
& \text{ENED} \\
8\left[ 1+{\ln \left( \frac{1+\Gamma _{+}}{2}\right) }-\Gamma _{+}\right] ,
& \text{LNED}%
\end{array}%
\right. ,$$$$\Gamma _{+}={\sqrt{{1}+\frac{{q^{2}}}{{\beta ^{2}}{r_{+}^{2(n-1)}}}},}$$ which shows that the temperature depends on the Lovelock parameter as well as nonlinearity factor of electrodynamics.

Now, we calculate the entropy of the black hole solutions. Since we regard the Lovelock gravity, the area law of the Black hole entropy does not satisfy in general [@Entropyfail]. The expression of the entropy of Lovelock black holes may be derived 