  We study the metric and topological properties of the space $\mathscr{D}(G)$
of left-invariant hyperbolic pseudometrics on the non-elementary hyperbolic
group $G$ that are quasi-isometric to a word metric, up to rough similarity.
This space naturally contains the Teichm\"uller space in case $G$ is a surface
group and the Culler-Vogtmann outer space when $G$ is a free group. Endowed
with a natural metric reminiscent of the (symmetrized) Thurston's metric on
Teichm\"uller space, we prove that $\mathscr{D}(G)$ is an unbounded
contractible metric space and that $\mathrm{Out}(G)$ acts metrically properly
by isometries on it. If we restrict ourselves to the subspace
$\mathscr{D}_{\delta}(G)$ of the points represented by $\delta$-hyperbolic
metrics with critical exponent 1, we prove that it is either empty or proper.
We also prove continuity of the Bowen-Margulis map from
$\mathscr{D}_{\delta}(G)$ into the space $\mathbb{P}\mathcal{C}urr(G)$ of
projective geodesic currents on $G$, extending similar results for surface and
free groups, and the continuity of the (normalized) mean distortion as a
function on $\mathscr{D}(G)\times \mathscr{D}(G)$.
