The magnetodynamical properties of nanometer-thick yttrium iron garnet films are studied using ferromagnetic resonance as a function of temperature. The films were grown on gadolinium gallium garnet substrates by pulsed laser deposition. First, we found that the damping coefficient increases as the temperature increases for different film thicknesses. Second, we found two different dependencies of the damping on film thickness: at room temperature, the damping coefficient increases as the film thickness decreases, while at T = 8 K, we find the damping to depend only weakly on the thickness. We attribute this behavior to an enhancement of the relaxation of the magnetization by impurities or defects at the surfaces.