I realize now that my post may not have been very clear. What I meant was, yeah, when you increase load you will see higher power draw. The reason for this is because there are more circuits active. Modern CMOS circuits, especially the modular ones like processors, use transistors to switch off chunk of circuitry at low loads.
But for a given circuit the power draw varies with frequency. What happens when you increase the load is more circuit elements switch on, so while you had 10 % of the chip drawing power before, you may now have 50% of it drawing power.
So that's a different cause altogether from what happens when frequency is varied. But for any given circuit in the on state (i.e. not power-gated or switched off), increasing frequency will increase power draw.