If $\deg(L)\leq 0$, then it is clear from the definition that $\vol(L,\|\ \|_{1},\|\ \|_{2})=0$. So we may assume that $L$ is ample. We need the energy $E(L,{\|\ \|}_{1},{\|\ \|}_{2})$ with respect to continuous semipositive metrics ${\|\ \|}_{1},{\|\ \|}_{2}$ on $L^{{\mathrm{an}}}$ introduced in [BGJKM16, Def. 2.4.4]. By Corollary 3.11, the envelopes ${P}({\|\ \|}_{1})$ and ${P}({\|\ \|}_{2})$ are semipositive model metrics on $L^{{\mathrm{an}}}$. In particular, they are continuous and hence it follows from [BGJKM16, Cor. 6.2.2] that

$\displaystyle\vol(L,{\|\ \|}_{1},{\|\ \|}_{2})=E(L,{P}({\|\ \|}_{1}),{P}({\|\ \|}_{2})).$

In the case of semipositive model metrics associated to line bundles $\mathscr{L}_{1},\mathscr{L}_{2}$ on a ${K^{\circ}}$-model $\mathscr{X}$, our assumption $\Gamma\subset\mathbb{Q}$ yields that the energy is defined as a $\mathbb{Q}$-linear combination of intersection numbers of the line bundles $\mathscr{L}_{1},\mathscr{L}_{2}$ on $\mathscr{X}$ proving the claim. ∎