Not really. Heat is transferred away from the surface of the bore by conduction into the barrel walls and finally to the outer surface of the barrel, which is then cooled by air (mostly convection) at a much slower rate. Heat transfer occurs at a faster rate for a higher temperature difference. More mass to absorb heat means that a higher temperature gradient, and thus faster heat transfer, can be maintained for a longer period of time.
My prediction is that the bore temperatures would be roughly the same for a given outer temperature for both barrels.
In order to visualize this, think of the barrel as being made up of a large number of very thin layers. Heat is absorbed by the first layer from the bore, then the next one, and so on, until it reaches the outside. The heavier barrel has more layers to heat and can therefore fire more shots before a given outer temperature is reached or have a lower outer temperature for the same number of shots.
