  Let $H$ be a hyperbolic group, $A$ and $B$ be subgroups of $H$, and
$gr(H,A,B)$ be the growth function of the double cosets $AhB, h \in H$. We
prove that the behavior of $gr(H,A,B)$ splits into two different cases. If $A$
and $B$ are not quasiconvex, we obtain that every growth function of a finitely
presented group can appear as $gr(H,A,B)$. We can even take $A=B$. In contrast,
for quasiconvex subgroups A and B of infinite index, $gr(H,A,B)$ is
exponential. Moreover, there exists a constant $\lambda > 0$, such that
$gr(H,A,B)(r) >\lambda f_H(r)$ for all big enough $r$, where $f_H(r)$ is the
growth function of the group $H$. So, we have a clear dychotomy between the
quasiconvex and non-quasiconvex case.
