Sounds like a temperature issue. Clean your fans and make sure they're working ok, then use something like CPUID's HWMonitor to keep an eye on the temperature. Otherwise you may have some hardware component that is beginning to fail and which is caught out by the demands FFXIV makes on it - eg PSU, memory chips, graphics card etc.

Those sorts of problems will be exacerbated by the fact that you're playing on a laptop, of course.