If you have well water I would hire a pro because you may have too high of sulphur content that would be reduced in higher temperatures. Also you might want to check and see if air could be getting into your pipes.
This is avery interseting question as the water if it is smelly should be smelly everywhere. If the bathroom sink is not cleaned properly then it may be smelly but there is no reason for the water to be smelly. You check the point from where the water is supplied to your bathroom.
