Get one o those air mattress inflating devices which is powered by four D-cells. then get some flexible tubing of the right size. Use this to insert the tubing in the action and blow air up the barrel.
I find that I can cool my .300 Weatherby, after firing three shots, to the same extent with this device after 5 minutes that I can in10 minutes just sitting it down muzzle up.
For some reason, my .300 Weatherby barrel gets A LOT hotter than my .375 H&H after firing three shots. The Wby uses about 10 grains more powder and I'm sure the pressures higher.
