  For any non-elementary hyperbolic group $\Gamma$, we find an outer
automorphism invariant geodesic bicombing for the space of metric structures on
$\Gamma$ equipped with a symmetrized version of the Thurston metric on
Techim\"uller space. We construct and study a boundary for this space and show
that it contains many well-known pseudo metrics including those coming from
actions on $\text{CAT}(0)$ cube complexes, real trees and coned-off Cayley
graphs. As corollaries we deduce length spectrum rigidity results, regularity
results for Manhattan curves, optimal growth rate results for Anosov
representations and results regarding continuous extensions of translation
distance functions to the space of geodesic currents. Using our results for
geodesic currents we settle a conjecture of Bonahon in the negative.
