With this knowledge you can warm your cold hands.
Yes, breathing carries oxygen through your blood to your muscles and widens your blood vessels so blood can flow properly. You exhale the oxygen that you do not use as the waste material carbon dioxide. If you breathe too fast, you exhale the oxygen as carbon before it is used up by your blood vessels. As a result, your blood vessels do not open far enough, so that your blood flow deteriorates. As a result, you get colder. That only applies if you consciously breathe quickly, hyperventilating does not cool you down.
The reverse also works
Cold hands? Breathe in and out slowly for three minutes. After exhaling, wait as long as possible before inhaling again. This allows carbon dioxide to flow back into your blood. This opens your vessels further and you feel that your hands are slowly getting warmer.