Based on the Boltzmann transport equation, we demonstrate that the thermal conductance per unit width of a sufficiently thin polar nanofilm supporting the propagation of surface-phonon polaritons along its surfaces is independent of the material properties and is given by 12z(3)kB3T2/ch2, where kB and h are the respective Boltzmann and Planck constants, while c is the light speed in vacuum, T is the temperature, and z(3) is the Riemann zeta function. The huge propagation length of these energy carriers establishes that this quantization holds not only for temperatures much smaller than 1 K, as is the case of electrons and phonons, but also for those comparable to room temperature, which can significantly facilitate its observation and implementation in the thermal management of nanoscale electronics and photonics.