This "low power mode" is a myth, the brain pulls a pretty constant amount of power (about 20% of metabolic energy), and most power isn't consumed by the action potential/neurons firing, but rather in keeping the neuron ready to fire.

And the regions involved in the default mode network are actually some of the metabolically most active regions in the brain, involved in memory consolidation, future simulation, theory of mind, and self-narrative; among other things.