  We study the thermal transport properties of general conformal field theories
(CFTs) on curved spacetimes in the leading order viscous hydrodynamic limit. At
the level of linear response, we show that the thermal transport is governed by
a system of forced linearised Navier-Stokes equations on a curved space. Our
setup includes CFTs in flat spacetime that have been deformed by spatially
dependent and periodic local temperature variations or strains that have been
applied to the CFT, and hence is relevant to CFTs arising in condensed matter
systems at zero charge density. We provide specific examples of deformations
which lead to thermal backflow driven by a DC source: that is, the thermal
currents locally flow in the opposite direction to the applied DC thermal
source. We also consider thermal transport for relativistic quantum field
theories that are not conformally invariant.
