Usually a sure sign of a bbl heating up will be VERTICAL stringing. This,depending on which other parts of the gun (forearm,etc) are touching the bbl and how hard (screwed to,clamped to,etc,etc).
When shooting some of my precission guns,I actually take a small fan to the range with me. It is blowing on the bbl during the entire shooting session.
For cooling and for bbl harmonics is the reasons that target guns,and the better hunting guns have a floated bbl. --- SAWMAN