  For a Riemannian manifold $(M,g)$ which is isometric to the Euclidean space
outside of a compact set, and whose trapped set has Liouville measure zero, we
prove Weyl type asymptotics for the scattering phase with remainder depending
on the classical escape rate and the maximal expansion rate. For Axiom A
geodesic flows, this gives a polynomial improvement over the known remainders.
We also show that the remainder can be bounded above by the number of
resonances in some neighbourhoods of the real axis, and provide similar
asymptotics for hyperbolic quotients using the Selberg zeta function.
