How do I keep PCBs cool when soldering?
Shouldn't need to cool them while hand soldering. In fact, cooling them whilst soldering would induce a lot of thermal stress with cool areas of the PCB and really hot areas in other bits of the PCB.

You may be overheating the PCB with incorrect soldering technique or maybe your PCB has big ground polygons/planes and you're soldering on joints connecting to these ground polygons/planes without a thermal relief.