Usually it will control the hot water. If the temperature starts rising it will shut down the hot water, and if there is not enough cold water to balance it, it will eventually shut the hot water off entirely. If there is no hot water, it would try to keep adding more, but the water would just stay cold. Most have a paraffin "motor" and it operates fairly rapidly, but NOT instantaneously. Most do not have "room" for both pressure balance and thermostats, and a pressure balancing feature would be redundant.