Ayep - at the very most it might get you a 12-24 hour head start, but since most HP recipes are full water, they can actually take longer to cure.
All heating does is speed up the saponification reaction a bit - when it cools down it may be all soap, and technically safe to use, but it is nowhere near cured, and not nearly as nice as it will be with a proper cure.
Cure is what happens AFTER the saponification reaction has finished, and makes the soap lather better, last longer when in use, and increases the gentleness and conditioning of the soap.
Nothing replaces cure but time. Try it on your own soaps and see the improvement.
CPOP and HP methods typically shorten the time to saponify the soap. After saponification is over, cure time begins.
The rule of thumb for most people is to allow at least 4 weeks for curing, regardless of how the soap is saponified. If you only think of cure as time the soap needs to dry down to a slow rate of moisture loss, then you really want to look at how much moisture is in the soap when the cure time starts, not how the soap was saponified. Soaps starting with more water will take longer to get to a stable moisture content. This is true whether you use HP, CPOP, regular CP methods, etc. But even after the soap is dry enough, it often continues to change in its quality of lather, longevity, etc.
Try it yourself -- test your soaps at various times during a month-long cure and evaluate the quality and amount of lather, overall hardness, skin feel after washing, and general appearance of the soap. If you can keep going, test the same soaps once every few months for a year and see if and how they change.