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It is not so much keeping the barrel cool, it is minimizing the harmonic barrel vibrations caused by the bullet moving through the bore.
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A given barreled action vibrates at the same frequency and amplitude for every shot fired with a given load. The only thing you can change is the amplitued by using more or less powder. If you want to get minimum amplitudes in a 30 caliber barreled action's vibrations, chamber it for the .30 Luger round; that's the smallest 30 caliber cartridge I know of.
What's more important is where the bullet leaves in the muzzle axis vertical swing through a small arc. Bullets leaving faster will leave at a different angle than slower ones. If all the bullets leave on the muzzle axis up swing just before it's at its peak, slower ones will leave at a higher angle than faster ones and their trajectories will compromise so they both hit the same place on the target; the ideal and most accurate situation.
No properly made barrel correctly fit to a receiver will change point of impact as its heated up. Any barrel that does change point of impact going from ambient to very hot has poor metal properties or is fit to the receiver without the receiver's face being squared up with the barrel's chamber and threaded tenon axis. Match rifles so built will shoot dozens of shots to the same place as they go from cold to skin-burning hot. Few, if any, factory sporting rifles are so fit, hence, their typical issues of walking shots on target as their barrels heat up. Have your factory barrel refit to its factory receiver by facing the receiver properly then putting in a thin shim between the barrel and receiver so it clocks in the same maintaining the correct headspace and the shot-walking issues typically go away providing the barrel's totally free floating in the first place.
Best accuracy with any bullet happens when it's a bit larger than the barrel's groove diameter. At least .0003" larger but it can be as much as .0015" larger and still shoot no worse than 1/2 MOA at 600 yards. If your barrel's groove diameter is larger than the bullets you shoot, really good accuracy is mostly a dream.