In this paper, we propose and study a numerical scheme for the Navier-Stokes-Fourier system derived and studied by the authors in Boittin at al. (2023). This system models hydrostatic free surface flows with density variations depending on temperature or salinity. We show that the finite volume/finite element scheme presented -- based on a layer averaged formulation of the model -- is well-balanced with regards to the steady state of the lake at rest and preserves the nonnegativity of the water height. A maximum principle on the density is also proved as well as a discrete entropy inequality (when the thermal and viscous effects are neglected). Some numerical validations are finally shown with comparisons to 3D analytical solutions and experiments.