If you do find that the thin barrel needs cooling, you might try the fan built into a chamber flag that is available from barrelcool.com.
http://barrelcool.com/product-category/barrelcool/

I bought one for about $36 about 18 months ago, if I remember correctly.
It has a tube the fits into the chamber and I find it cuts cooling time in half over just letting the rifle sit.
That really makes a difference when the temperature on the firing line is 85 degrees.

I have a LCD strip on the barrels of my target rifles so I can tell when the barrel temp is getting too high.
When a bull barrel gets to around 122 deg F. I find that the POI drops about 1/4 inch.
My 6.5mm Creedmoor 12 LRP groups really well and 1/4 inch is a 20% increase in group size with a good load.