  We prove that that for all $\eps$, having cogrowth exponent at most
$1/2+\eps$ (in base $2m-1$ with $m$ the number of generators) is a generic
property of groups in the density model of random groups. This generalizes a
theorem of Grigorchuk and Champetier. More generally we show that the cogrowth
of a random quotient of a torsion-free hyperbolic group stays close to that of
this group.
  This proves in particular that the spectral gap of a generic group is as
large as it can be.
