  In the present paper we consider a Gaussian thermostat on a compact
Riemannian surface with negative thermostat curvature. In the case of surfaces
with boundary, we show that the thermostat ray transform with attenuation given
by a general connection and Higgs field is injective, modulo the natural
obstruction, for tensors. We also prove that the connection and Higgs field can
be determined, up to a gauge transformation, from the knowledge of the parallel
transport between boundary points along all possible thermostat geodesics. In
the case of closed surfaces, we obtain similar results for the ray transform
with some additional conditions on the connection. Under the same condition, we
study connections and Higgs fields whose parallel transport along periodic
thermostat geodesics coincides with the ones for the flat connection.
