It's not mold, but is rancidity. But bashful is quite mistaken that it happens in cold process and not in hot process. And even beyond that, she is mistaken that it appears after a few weeks.
Even if she were correct, It's not likely that it's the heat killing the mold or its spores. While there ARE molds which can survive very high pH systems, it's unlikely that something that can survive the lye solution that is reacted with the oils will then be impacted by the potential slight difference in temperature that the mixture may experience in CP vs HP (CP still heats up, and gels, as the saponification is exothermic, in HP heat is added to speed up the initial reaction.
What is more likely is that bashful (who isn't very) is changing other variables in her soapmaking. Different vessels, perhaps, or other equipment or other things.