We use the lubrication approximation to investigate the steady flow of a thin rivulet of viscous fluid with prescribed volume flux draining down a planar or slowly varying substrate that is either uniformly hotter or uniformly colder than the surrounding atmosphere, when the surface tension of the fluid varies linearly with temperature. Utilizing the (implicit) solution of the governing ordinary differential equation that emerges, we undertake a comprehensive asymptotic and numerical analysis of the flow. In particular it is shown that the variation in surface tension drives a transverse flow that causes the fluid particles to spiral down the rivulet in helical vortices (which are absent in the corresponding isothermal problem). We find that a single continuous rivulet can run from the top to the bottom of a large horizontal circular cylinder provided that the cylinder is either warmer or significantly cooler than the surrounding atmosphere, but if it is only slightly cooler then a continuous rivulet is possible only for a sufficiently small flux (though a rivulet with a discontinuity in the free surface is possible for larger values of the flux). Moreover, near the top of the cylinder the rivulet has finite depth but infinite width, whereas near the bottom of the cylinder it has finite width and infinite depth if the cylinder is heated or slightly cooled, but has infinite width and finite depth if the cylinder is significantly cooled.