For a compact Riemannian manifold of negative curvature, the geodesic foliation of its unit tangent bundle is independent of the negatively curved metric, up to Hölder bicontinuous homeomorphism. However, the Riemannian metric defines a natural transverse measure to this foliation, the Liouville transverse measure, which does depend on the metric. For a surface S, we show that the map which to a hyperbolic metric on S associates its Liouville transverse measure is differentiable, in an appropriate sense. Its tangent map is valued in the space of transverse Hölder distributions for the geodesic foliation.