Yes, they can. Hot water opens up the hair cuticle possibly making more color leach out. In contrast, cold water tends to 'seal' the cuticle. Be aware that the more time you rinse colored hair under hot water, the more it can lose its vibrancy. Therefore, giving the hair a quick blast of cold water or placing a shower cap over your hair after it's been washed are two much easier (and less chilly) ways to protect colored hair.