In addition to the other answers, the tub in my old house the water DID drain out from the shower head when you turned it off, you could hear it running down and putting your hand over the shower head would stop it.
But the water was still cooler when it came out, and I finally concluded that was because the empty pipe and shower head would have cooled off, and the hot water going up the pipe was getting cooled off by contact with the metal.
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