You’re presumably thinking of the “Affinity Laws”, which, according to Wikipedia (and plenty of other sources), “apply to pumps, fans, and hydraulic turbines. In these rotary implements, the affinity laws apply both to centrifugal and axial flows.”
This is, IMO, one of the worst kinds of science writing. Wikipedia, and plenty of other sources, make little mention of when the do and don’t apply or, relatedly, why they’re true and why they can’t always be true.
They generally apply to situations where a pump is pumping fluid through something like a filter or a long pipe where the pipe is a closed loop or at least the ends are at the same elevation (e.g. a swimming pool pump, except when pumping from a pool into a higher hot tub). So you have no actual work being done by moving fluid, and you can run the pump slower, and thus move less fluid per unit time, thus reducing friction in a manner that the pressure that the pump needs to overcome goes all the way to zero as the flow rate approaches zero.
But the affinity laws are not really anything fundamental about pumps, and they certainly do not override conservation of energy.
Now consider a refrigerator. The compressor is pumping refrigerant from an (approximately) fixed low pressure to a fixed high pressure. (The fluid goes back from high pressure to low pressure via a capillary tube or expansion valve or similar lossy device -- it gets its pressure increased in the gas phase and decreased in the liquid phase.) There's some friction, but after subtracting friction, the pressure is independent of flow rate, and thus the work done per unit flow is independent of flow rate, and the pump power scales linearly with flow as opposed to super-linearly as the affinity laws suggest.
Also, the compressor is a positive-displacement pump, and the affinity laws don't even pretend to apply to these.
(A well pump is another common system where the affinity laws will lead to nonsensical results. If you want to size a well pump properly, you need to know the height that you're raising the water, the output pressure you need, and the range of flows that you want. And then you look at the actual measured performance curves of the pumps (and their drives) that you are considering, and you pick something appropriate.)
All that being said, variable-speed fridges exist, and they're kind of nice in that they try to run continuously and quietly instead of alternating between full-power (and loud) and all the way off. And they are probably a bit more efficient because there's less friction and because the motors are likely to be more efficient three-phase designs instead of the not-actually-amazing single-phase motors you'll find in older fridges.