A nonlinear magnon interaction has been investigated in an yttrium iron garnet/Pt film. In the bilayer film, multiple microwave absorption signals due to the excitation of magnetostatic spin waves, including magnoeto-static surface waves and backward volume magnetostatic spin waves, were observed at room temperature. The microwave absorption spectrum was found to be changed drastically by decreasing temperature. This drastic change in the microwave absorption spectrum is attributed to the appearance of the three-magnon splitting process; the spin-wave dispersion calculated by taking into account the temperature variation of the saturation magnetization indicates that the change in the microwave absorption spectrum appears when the lowest frequency of the spin-wave dispersion becomes lower than a half of the microwave excitation frequency. At this condition, the energy and momentum conservation laws can be fulfilled in the process where a pumped magnon splits into two magnons. These results demonstrate that the nonlinear magnon interaction can be controlled not only by excitation frequency but also by temperature.