Amorphous solids yield at a critical value Sigma(c) of the imposed stress Sigma through a dynamical phase transition. While sharp in athermal systems, the presence of thermal fluctuations leads to the rounding of the transition and thermally activated flow even below Sigma(c). Here we study the steady-state thermal flow of amorphous solids using a mesoscopic elastoplastic model. In the Hebraud-Lequex (HL) model we provide an analytical solution of the thermally activated flow at low temperature. We then propose a general scaling law that also describes the transition rounding. Finally, we find that the scaling law holds in numerical simulations of the HL model, a two-dimensional (2D) elastoplastic model, and previously published molecular dynamics simulations of 2D Lennard-Jones glass.