Lets start first with cold inflation - hopefully you know the exact weights of each wheel position, if not each position at least the weight of each axle position. Once you determine those (keep in mind that these weights are as you would normally be packed at including any water you may carry) then you can set the tires to the proper pressure. Just because the sidewall states 125 PSI, you may only require say 90 PSI to safely carry your weight. Over inflation can create irregular wear issues. There are a number of sights that can provide you with PSI and weights for your particular tire size. ALWAYS SET ALL OF THE TIRES TO THE HIGHEST PSI THAT MATCHES THE HEAVIEST WHEEL/AXLE POSITION.
Next, as for the changes in tire pressure - yes the tires will increase in PSI depending on a number of factors, load, speed, and inflation to name a few. Ambient temp can and will also pay a role in any increase (along with variance in altitude). Keep one thing in mind, cold inflation is the inside of the tire matching the exterior in terms of temperature. As you check “hot” inflation NEVER deflate the tire if it appears to be higher then the recommended pressure.
If if you set your pressure and you see variations with the TPMS I would double check with an air gauge to verify that there are truly differences between the wheel positions. There is a possibility that the TPMS could be off.
Finally, your temp difference (assuming everything started out correct) could indicate more weight on the rear axle then the front axle. Keep in mind I’m only going off your info.