Taking a hot shower can make your temperature go up but only for a while after you get out of the shower.
Your body temperature may increase by 5 to 10 F degrees during a hot shower but the body temperature will return to normal after you get out of the shower and cool off.
So the body temperature increase from a hot shower is only temporary.
Having a hot shower or hot bath can increase your temperature but after you get out of the hot shower or hot bath your temperature will usually return to normal within 20 to 30 minutes or so.
Taking a hot shower or hot bath can be good for the flu, cold etc as it helps raise your body temperature which helps to fight off infections.
The hot bath or hot shower is also good for stuffy nose and congestion as well.
A hot shower can raise your temperature a little bit and then after the show your body temperature should return to the normal body temperature.
The water hitting your skin is hot coming from the shower so it can naturally increase your bodies temperature but it should affect your body temperature for too long.
Your core body temperature will return to normal after you cool down from the shower.
Also when you're taking a hot shower your blood vessels become violated and your blood flow increases which leads to the increase in body temperature and red looking skin.