It will not hurt a healthy person to take a hot or a cold shower, but either one can increase your heart rate. Another factor that can add to this is worry about getting to work on time and the problems you have to deal with when you get there. Hurry makes you move faster, and this too can increase your heart rate.
A hot shower will raise your body temperature. If it goes much above your normal temperature, your hypothalamus will act much the same as it would if you had a high fever. Your pores will open, you will sweat and your heart will speed up to quickly bring blood close to the surface of the skin to cool it. Your body is designed to protect the brain and the vital organs from excess heat regardless of the source. With heat exhaustion, the pulse will become rapid and weak.
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A cold shower can wake you up and make you feel energetic. The message sent to your hypothalamus is that you are too cold. Again, to protect the brain and vital organs, it takes action. The pores close, goose bumps may form and the heart speeds up to move warm blood more quickly. You may start to shiver. Shivering is an involuntary muscle action that uses the heat produced by muscle activity to warm your body. With actual hypothermia, the heart rate slows.
The increased heart rate may make you uncomfortable, or your doctor may want you to avoid this stress on the heart or other organs. A shower just slightly above body temperature will not trigger the body to react. The room should also be comfortably warm. Try to not hurry and take time to dry yourself thoroughly. Dress warmly before you leave the room if the rest of the house is chilly.
You can do quite a few exercises in the shower. They may also increase your heart rate slightly, but much less than aerobics. Try sucking your belly in hard, holding it for 10 seconds and then pushing it out again. Repeat as time permits. Tighten and then relax the muscles in your pelvic floor to strengthen them. Many muscle groups can be exercised this way. Do be careful and limit exercises to ones that can be performed safely on a slippery surface.