I think your friend is over-thinking it. According to this study, there are several factors that contribute to continued burning of calories after exercise, one of which is:
Decrease in Body Temperature: As energy is liberated from the
exercising muscle tissues of the body, heat is produced. Thus, during
EPOC, the body must expend energy to return to the normal core body
So theoretically, if you are taking a hot shower, you are actually causing your body to work harder to get back down to the normal core temp, which should result in more calories burned. Taking a cold shower should have the opposite effect. If you consider that your body should be back to normal temp about 20 minutes after your workout anyways, and assuming you are cooling down normally (walking for 5 - 10 minutes after your workout), the differences in calories burned wouldn't be noticeable based on the temperature of your shower.