I assume you have a pre 81 BLR... easiest way to tell is the magazine is not flush with the bottom of the receiver.
I can believe the reasonably light barrel heats up after 5 shots. With mine I use 3 shot groups. I find that if I blow the firts shot and the deer is running I have a very small chance of connecting with a second shot. I have yet to get a chance to unload the rifle at one animal while hunting.