For a generic metric on an $m$-dimensional the manifold the eigenvalues of the Laplacian are all simple. Fix such a metric and denote the coresponding eigenvalues by
$$ \lambda_1, \lambda_2,\cdots $$
Using Weyl's asymptotic expansion we conclude that
$$\lambda_n\sim const . n^{m/2}. $$
Thus for any polynomial $P$ of degree $d>m/2$ we have
$$ \lim_{n\to\infty} \lambda_n/P(n) = 0. $$
