Not necessarily. The best way to make your hair color last longer is by not washing your hair every single day. It can be a hard transition to make if you’re used to shampooing every day, but it really does offer the most hair color protection and is the best thing you can do to lengthen the life span of single processes and semi-permanent color. Try dry shampoos to soak up oil and extend the life of your style between shampoos.

What cold water will do is help make your hair shinier. I like to treat hair like skin: Just like washing and moisturizing your skin is important, so is washing and conditioning your hair the right way. A cool rinse closes your pores and tightens your skin; a cool rinse on your hair closes your hair cuticles, tightening them up. When the surface of the hair is smooth, it reflects more light, making your hair much shinier. Of course, even though I always try to do a cool rinse, some of my clients will jump out of the chair if the water is too cold -- in a shower, it’s even trickier! 

Hot water would have to be near scalding for it to damage your hair so badly it strips the color. (Moreover, extra-hot water is bad for your scalp and skin!) But chances are, you’re not showering in water that hot.