of electrodynamics [@SmarrNew].

Geometrical techniques (scaling argument) were used to derive an extension of the first law and its related modified Smarr formula that includes variations in the cosmological constant, Lovelock coefficient and also nonlinearity parameter. In our case, Lovelock gravity in the presence of the NED, $M$ should be the function of entropy, pressure, charge, Lovelock parameter and BI coupling coefficient [SmarrNew]{}. Regarding the previous section, we find that those thermodynamic quantities satisfy the following differential form $$dM=TdS+\Phi dQ+VdP+\mathcal{A}_{1}^{\prime }d\alpha _{2}+\mathcal{A}%
_{2}^{\prime }d\alpha _{3}+\mathcal{B}d\beta .  \label{GenFirstLaw}$$where we have achieved $T$ and $\Phi $, and one can obtain$$\begin{aligned}
V &=&\left( \frac{\partial M}{\partial P}\right) _{S,Q,\alpha _{2},\alpha
_{3},\beta }, \\
\mathcal{A}_{1}^{\prime } &=&\left( \frac{\partial M}{\partial \alpha _{2}}%
\right) _{S,Q,P,\alpha _{3},\beta }, \\
\mathcal{A}_{2}^{\prime } &=&\left( \frac{\partial M}{\partial \alpha _{3}}%
\right) _{S,Q,P,\alpha _{2},\beta }, \\
\mathcal{B} &=&\left( \frac{\partial M}{\partial \beta }\right)
_{S,Q,P,\alpha _{2},\alpha _{3}}.\end{aligned}$$

Using the redefinition of $\alpha _{2}$ and $\alpha _{3}$ with respect to the single parameter, $\alpha $, we can rewrite $\mathcal{A}_{1}^{\prime
}d\alpha _{2}+\mathcal{A}_{2}^{\prime }d\alpha _{3}$ as a single differential form $$\begin{aligned}
d{\alpha _{2}} &=&\frac{1}{{(n-2)(n-3)}}d\alpha , \\
d\alpha _{3} &=&\frac{2{\alpha }}{{3(n-2)(n-3)(n-4)(n-5)}}{d\alpha .}\end{aligned}$$

Moreover, by scaling argument, we can obtain the generalized Smarr relation for our asymptotically adS solutions in the 