Yes, it takes more power, but only on bass notes in that lower, boosted range, and in real music, this is just momentary.

Are there any drawbacks to doing this?

In a way, your Pi folded horn does the same thing, but you are boosting the low end via the horn itself, instead of with extra power of the amp

This would be the so-called EBS alignment, where the tuning frequency is set very low and peaked so that a shelf is formed. It provides flat response down to fh, and then response is reduced but relatively flat below fh down to fl. That's why it requires increased power and is why guys today call it an "Extended Bass Response" or EBS alignment. Apparently Electrovoice called it "Step Down" and I called it "EBS Stepped Response from ported cabinet with specific peaking." This and other characteristic response curves are shown in a post called "Response curves of closed vs. vented systems."

I don't know if you saw the posts, but "Romy the Cat" uses a system like you described. You know the one with the red or blue metal-flake plastic radial horns.

Dave

Adam