Toyota Hybrid system does not have the side belt, so heating and cooling are not directly related to the engine.
In classic engine, coolant pump is powered by engine directly, so coolant goes different routes based on it's temperature (because you can't stop the pump). Hybrid system is more effective because of electric heat pump that is operated by a microcontroller (complete stop, full stream and anything between). That's why hybrid engine gets heated much faster and can send the heat to the cabin faster and more efficient.
For cooling, again, no direct connection to the engine anymore. Classic car design connects ac compressor to the engine, so bigger rpm means cooler air. In hybrid you can clearly see the orange wires to the ac compressor. It has all its power with or without the gas engine turned on.
Overall, climate system of hybrid and non-hybrid are apples to oranges. IMHO, hybrid climate performs better and more efficient (and that's one of the reasons why I got one).