Your second thought is the answer. It's all about the load and how the regulator deals with it. As the demand goes down, so does heat and less of a power demand upon the engine.
EDIT: (To provide more context.)
To put more emphasis upon what I'm saying, I'll reference this article which talks about how the alternator and regulator work:
Lights, ignition, and accessories use power from the electrical system. Every time we switch an accessory ON, more power is drawn from the system. Voltage (electrical pressure) drops as power is drawn from the system, and then the voltage regulator causes the alternator to make more current.
If you were going in the opposite direction of what you're asking and drawing more from the system, the voltage regulator makes the alternator work harder. Therefore, if you demand less of a draw on the system, the regulator isn't going to tell the alternator to kick in as often to do its job.
In the electrical system, the voltage regulator “turns the alternator ON,” or “turns OFF the alternator” as needed to maintain voltage at the proper level. And in the air compressor system the pressure regulator stops and starts the compressor as needed to maintain the proper level of pressure.
The useful electrical system will require an alternator that can produce an average of more output than we use, and the regulator will limit system voltage to the safe level we need. ... Alternators make heat as a by-product of making electrical power, and the more power they supply the more heat they make. Some models of alternators can stand to put out a much higher percentage of their gross output rating than others, during extended periods of operation.
Since we know that increased load on an alternator will make heat, the bi-product of drawing less from the system is the alternator will run cooler because it's not having to generate as much power over a given amount of time.