The damping of spin waves transmitted through a two-port magnonic device implemented on a yttrium iron garnet thin film is shown to be proportional to the temperature gradient imposed on the device. The sign of the damping depends on the relative orientation of the magnetic field, the wave vector, and the temperature gradient. The observations are accounted for qualitatively and quantitatively by using an extension of the variational principle that leads to the Landau-Lifshitz equation. All parameters of the model can be obtained by independent measurements.