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Disclaimer: I am not an audio engineer or musician, I have a college-level understanding of physics, and I dick around with video games and electronics as a hobby, so I have a little specialized understanding of audio, waves, energy, etc.
Human voices don't work like loudspeakers where the diaphragm vibrates. Our diaphragm moves air through constant pressure, and the larynx is what actually creates the vibration. A deep vs high-pitched voice is defined by the geometry in the larynx. It's more like a breath-operated instrument like brass or woodwinds, and in those, the difference between high and low notes isn't all that significant.
It might be more significant between a bass and alto saxophone, but the sheer volume of the instrument itself plays a greater role. And usually they're played louder than a typical singer.
I don't know if there is actual measurement of energy expenditure for human vocalization. You could measure energy of the sound waves, but I'm not sure if that would directly correlate in the way you're thinking.
Edit: as mentioned I'm no engineer, so if the larynx uses any kind of resonance to produce its sound, I have no idea how that works.
total peanut gallery but when people talk about instruments that take a ton of breath to play it does seem to be the lower note ones like tuba. I don't even play instruments this mainly comes from friends who played complaints.
Tuba and flute take the most breath to play. You have to move air FAST through a flute, while a brass instrument is more about the vibration of your lips than sheer air going through.
ok yeah and now that I think of it flute was up there and its not known for low.