  We consider the Gaussian free field on $\mathbb{Z}^d$, $d$ greater or equal
to $3$, and prove that the critical density for percolation of its level sets
behaves like $1/d^{1 + o(1)}$ as $d$ tends to infinity. Our proof gives the
principal asymptotic behavior of the corresponding critical level $h_*(d)$.
Moreover, it shows that a related parameter $h_{**}(d) \geq h_*(d)$ introduced
by Rodriguez and Sznitman in arXiv:1202.5172 is in fact asymptotically
equivalent to $h_*(d)$.
