Just got my TST system cranked up and also just came off the road from a 920 mile run. Have 16's on the fifth wheel so I always start at 80 psi cold. It was generally around 56 degrees or cooler outside. One morning it was 40 outside and I noticed the tires generally had dropped from 80 to 78. The highest pressure indicated was 92 psi. Temp ran about 12 degrees higher than ambient.
I been wondering on this subject also so this was a good read.
I did not time it but sure was surprised how the tire temp and pressure quickly climbed close to the final running numbers when initially starting out.