Thermal switching provides an effective way for active heat flow control, which has recently attracted increasing attention in terms of nanoscale thermal management technologies. In magnetic and spintronic materials, the thermal conductivity depends on the magnetization configuration: this is the magneto-thermal resistance effect. Here we show that an epitaxial Cu/Co$_{50}$Fe$_{50}$ multilayer film exhibits giant magnetic-field-induced modulation of the cross-plane thermal conductivity. The magneto-thermal resistance ratio for the Cu/Co$_{50}$Fe$_{50}$ multilayer reaches 150% at room temperature, which is much larger than the previous record high. Although the ratio decreases with increasing the temperature, the giant magneto-thermal resistance effect of ~100% still appears up to 400 K. The magnetic field dependence of the thermal conductivity of the Cu/Co$_{50}$Fe$_{50}$ multilayer was observed to be about twice greater than that of the cross-plane electrical conductivity. The observation of the giant magneto-thermal resistance effect clarifies a potential of spintronic multilayers as thermal switching devices.