Ultrafast optical heating of the electrons in ferrimagnetic metals can result in all-optical switching (AOS) of the magnetization. Here we report quantitative measurements of the temperature rise of GdFeCo thin films during helicity-independent AOS. Critical switching fluences are obtained as a function of the initial temperature of the sample and for laser pulse durations from 55 fs to 15 ps. We conclude that nonequilibrium phenomena are necessary for helicity-independent AOS, although the peak electron temperature does not play a critical role. Pump-probe time-resolved experiments show that the switching time increases as the pulse duration increases, with 10 ps pulses resulting in switching times of ∼13 ps. These results raise new questions about the fundamental mechanism of helicity-independent AOS.