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We study the metric and topological properties of the space $\mathscr{D}(G)$of left-invariant hyperbolic pseudometrics on the non-elementary hyperbolicgroup $G$ 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 surfacegroup and the Culler-Vogtmann outer space when $G$ is a free group. Endowedwith a natural metric reminiscent of the (symmetrized) Thurston's metric onTeichm\"uller space, we prove that $\mathscr{D}(G)$ is an unboundedcontractible metric space and that $\mathrm{Out}(G)$ acts metrically properlyby isometries on it. If we restrict ourselves to the subspace$\mathscr{D}_{\delta}(G)$ of the points represented by $\delta$-hyperbolicmetrics 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 $\mathbb{P}\mathcal{C}urr(G)$ ofprojective geodesic currents on $G$, extending similar results for surface andfree groups, and the continuity of the (normalized) mean distortion as afunction on $\mathscr{D}(G)\times \mathscr{D}(G)$.

The space of metric structures on hyperbolic groups