I've always rinsed under hot water. I don't know if it improves the closeness of the shave, but it certainly improves my enjoyment. Don't much like cold stuff on my skin. I loathe the cold water rinse, but do it anyway!
I once tried warming my deo in my wife's microwave before we married. It was a very chilly South African morning. Ten seconds & it went bang! She was not pleased.