I used to have a 2009 RX450h. From memory, when you first start the car, it takes a few minutes before the heating kicks in. I believe this is to stop cold air from being blown into the cabin while the engine to comes up to running temperature. This is overriden if you select a high temperature.
Given that the temperature can be independently adjusted for right and left, I would assume that there are two thermostats, one for each side, but I could be wrong.
Not sure whether this is relevant to your question, but I seem to recall that the manual recommends that you should not select more than 3 degrees difference between the right and the left of the cabin. This aims to stop the two sides from 'competing' when one side tries to cool the car down, while the other tries to increase the temperature.