  We consider Anosov thermostats on a closed surface and the X-ray transform on
functions which are up to degree two in the velocities. We show that the
subspace where the X-ray transform fails to be s-injective is finite
dimensional. Furthermore, if the surface is negatively curved and the
thermostat is pure Gaussian (i.e. no magnetic field is present), then the X-ray
transform is s-injective.
