In \cite{Cha1}, the leading order term of the free energy of $\text{U(N)}$ lattice Yang-Mills theory in $Λ_n=\{0,\ldots,n\}^d\subset \mathbb{Z}^d$ was determined, for every $N\geq 1$ and $d\geq 2$. The formula is explicit apart from a contribution $K_d$ which corresponds to the limiting free energy of lattice Maxwell theory with boundary conditions induced by the axial gauge. By suitably adjusting the boundary conditions, we provide an equivalent characterization of $K_d$ that admits its explicit computation.