The question concerns the justification that Gromov gives for the proposition "if $V$ and $W$ are closed, negatively curved manifolds with isomorphic fundamental group, then their unit tangent bundles $S(V)$ and $S(W)$ are homeomorphic" (although it is open, or was at the time of writing, whether $V$ and $W$ are homeomorphic).
The proof is short. The fundamental group $\Gamma$ acts isometrically and properly discontinuously on the universal covers $\tilde{V}$ and $\tilde{W}$ (which are both homeomorphic to Euclidean space). By Milnor-Schwarz the ideal boundary $\partial_\infty X:=\partial_\infty \tilde{V}=\partial_\infty\tilde{W}$ and the action of $\Gamma$ on it depends only on $\Gamma$. Finally, the bundles $S(V)$ and $S(W)$ are the associated bundles to the $\Gamma$-principal bundles $\tilde{V}\to V$ and $\tilde{W}\to W$, respectively, with the same fiber $\partial_\infty X$. I don't see why the last observation implies that the total spaces of these associated bundles must be homeomorphic.
Do bundles associated to different classifying spaces (for $\Gamma$) with some fixed fiber $F$ (and fixed $\Gamma$ action on $F$) have homeomorphic total spaces? It sounds false to me.