  Although for photon Bose-Einstein condensates the main mechanism of the
observed photon-photon interaction has already been identified to be of
thermo-optic nature, its influence on the condensate dynamics is still unknown.
Here a mean-field description of this effect is derived, which consists of an
open-dissipative Schr\"odinger equation for the condensate wave function
coupled to a diffusion equation for the temperature of the dye solution. With
this system at hand, the lowest-lying collective modes of a harmonically
trapped photon Bose-Einstein condensate are calculated analytically via a
linear stability analysis. As a result, the collective frequencies and, thus,
the strength of the effective photon-photon interaction turn out to strongly
depend on the thermal diffusion in the cavity mirrors. In particular, a
breakdown of the Kohn theorem is predicted, i.e.~the frequency of the
centre-of-mass oscillation is reduced due to the thermo-optic photon-photon
interaction.
