A study has been performed on the behavior of the emissive characteristics of a cold graphite cathode in a magnetically insulated coaxial diode under the action of nanosecond accelerating pulses at a pulse repetition frequency (PRF) of up to 3.5 kHz. Emission was observed to degrade at PRF < 1 kHz and recover at PRF ~ 3.5 kHz. Estimates of the temperature conditions in the region of an explosive electron emission (EEE) center have shown that the pulse interval t ~1 ms suffices for this region to cool down to 300 K. The cooling occurs predominantly by heat conduction. For t ~ 0.3 ms, the residual heat is substantial. It has been proposed that there exists a frequency limit for the cathode microrelief polishing effect. The results of an experiment on studying the mechanism of cathode emission recovery with increasing PRF are presented. Micrographs of the cathode taken after aging, photographs of the cathode in operation, and analyses of the fractional composition of the material removed from the cathode suggest that an important role in the recovery of emission is played by the heating of some regions of the cathode emitting edge to the melting point of graphite. This counts in favor of the hypothesized dominant contribution of thermoelectron emission to the initiation of EEE due to the residual heat remaining in regions that have not cooled off during the pulse interval.