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A great deal of effort has been made over the last decades to develop a better polarized electron source for high energy physics. Several laboratories operate DC guns with a gallium arsenide photocathode, which yield a highly polarized electron beam. However, the beam's emittance might well be improved by using a superconducting radio frequency (SRF) electron gun, which delivers beams of a higher brightness than that from DC guns because the field gradient at the cathode is higher. SRF guns with metal and CsTe cathodes have been tested successfully. To produce polarized electrons, a Gallium-Arsenide photo-cathode must be used: an experiment to do so in a superconducting RF gun is under way at BNL. Since a bulk gallium arsenide (GaAs) photocathode is normal conducting, a problem arises from the heat load stemming from the cathode. We present our measurements of the electrical resistance of GaAs at cryogenic temperatures, a prediction of the heat load and verification by measuring the quality factor of the gun with and without the cathode at 2 K. We simulate heat generation and flow from the GaAs cathode using the ANSYS program. By following the findings with the heat load model, we designed and fabricated a new cathode holder (plug) to decrease the heat load from GaAs.