I know that it is common to say, "use hot water when washing your hands" or when you've got a cut, "wash your hands with warm water," etc. I was wondering, why is this the case? Since bacteria grow in warm environments, you would think that it might be beneficial for bacteria. Is this just another myth or is this actually true?
The bacteria wouldn't see any benefit from the warm water in the ~30-60 seconds you're washing your hands, neither would hot water "sterilize" your hands at temperatures you could tolerate. The reason you wash your hands with hot water is because the hot water+detergent soap mix is better for removing oil and dirt than cold water+detergent, which is supposedly where bacteria reside.
Interestingly, there was a study published saying that washing with hot water doesn't have much of an effect on bacterial load. Hot water for handwashing--where is the proof? (Laestadius and Dimberg, 2005). I'd be interested to look more into their methods but the paper is behind a paywall and I don't have access at home.
Also, this paper, Water temperature as a factor in handwashing efficacy by Michaels et al. is open and shows no evidence that hot water is any better than cold water for removal of microbes.
[W]ater temperature exhibits no effect on transient or resident bacterial reduction during normal handwashing with bland soap.
The goal of hand washing is to remove surface debris, including foreign pathogens. Most things that most people encounter on a daily basis dissolve more easily in hot water than in cold water. Thus, hot water around 100° F is used to facilitate this debris removal process, although the ideal culture temperature for E coli and a raft of related bacteria is around there too.
That being said, from the wikipedia on hand washing:
Contrary to popular belief however, scientific studies have shown that using warm water has no effect on reducing the microbial load on hands
It makes no difference at all. To sterilize your hands you would need to wash them with water at around 115 degrees C at 1.2 atmospheres of pressure for over 30 minutes – that's obviously not going to happen. The bacteria that normally reside on your skin can easily tolerate tap water, no matter whether the faucet is set to maximum hot or maximum cold, or just average. Even if the temperature of the water could appreciably slow down bacterial growth (which it can't), once you've washed your hands your skin temperature goes back to as it was before the washing in a minute or so.