  We prove that the rational cohomology $H^i(\mathcal{T}_g;\mathbf{Q})$ of the
moduli space of trigonal curves of genus $g$ is independent of $g$ in degree
$i<\lfloor g/4\rfloor.$ This makes possible to define the stable cohomology
ring as $H^\bullet(\mathcal{T}_g;\mathbf{Q})$ for a sufficiently large $g.$ We
also compute the stable cohomology ring, which turns out to be isomorphic to
the tautological ring. This is done by studying the embedding of trigonal
curves in Hirzebruch surfaces and using Gorinov-Vassiliev's method.
