This might not be considered an application, but Hilbert metrics have been studied geometrically and dynamically. Here are several examples of questions that have been partially or totally answered:

* when is a convex domain endowed with its Hilbert metric $\delta$-hyperbolic?

* what is the volume entropy of Hilbert metrics?

* does there exist convex sets that admit a cocompact group of isometries (relative to their Hilbert metric)? (the answer is yes!)

I guess that amoung these, the last part can be considered an application: Hilbert metrics yields interesting subgroups of $\mathrm{PSL}(n,\mathbb{R})$.

For more details you can look at the works of [Yves Benoist](http://www.math.u-psud.fr/~benoist/prepubli/prepublication.html) (in particular the "convexes divisibles" series and "Convexes hyperboliques et fonctions quasisymétriques", in french), [Constantin Vernicos](http://constantin.vernicos.org/Enfrancais/costiapub.html), [Ludovic marquis](http://perso.univ-rennes1.fr/ludovic.marquis/Recherche.html) and [Mickaël Crampon](http://mikl.crampon.free.fr/recherche.html).