The need for a volume (or surface area) normalization appears because the eigenvalues of the Laplacian scale inversely proportional to the area. More precisely, if you scale the metric as $\mu'=\gamma^2\mu$, then the Laplacian determinant scales with a factor $\gamma^{-\chi/3}$, with $\chi$ the Euler characteristic of the surface. So unless $\chi=0$, you need to normalize to unit area.
If I am not mistaken, the normalization actually ensures the additivity you are seeking, rather than spoiling it.