Yes, the each gap survives a small perturbation (even a Hoelder perturbation), see here. For smoothness of the Laplacian with respect to the metric see here.
But maybe, Weyl's asymptotic formula (see p155 of Chavel: Eigenvalues in Riemannian Geometry) $$(\lambda_k)^{d/2} \sim \frac{(2\pi)^d k}{\text{Vol}(D^d).\text{Vol}(M)}$$ (which holds for each compact Riemannian manifold of dimension $d$) is sufficient for you.