Controlling breathing rate is very important. I completed a couple of VO2 max tests last year. Test one, I went into an uncontrolled hyper breathing rate as I got to my maximum effort and lost control of the effort and the watts slipped. Test two, I controlled my breathing rate right through to maximum effort, no drop off in watts and scored significantly higher. Yes I was fitter second time around, but the effort did not decline. The exercise physiologist conducting the tests explained this phenomenon to me after the first test, and since then I have worked on controlling my breathing rate.
Another aside was an article I read in Runners World years ago about breathing through a straw building efficiency similar to training at altitude. I did this a lot during Covid when working remotely and doing 15min of breathing through the equivalent of a thickshake straw a couple of times a day. It seemed to be effective.
One more anecdote on breathing. During my daily commute of 9km to and from work on a relatively flat course, I work on “nose breathing” only. What this does is purposely slow me down to ride a true recovery pace. Once my HR is over 125 beats, I need to breathe through my mouth as well. This could also be applied to a recovery ride on the trainer.
I’m also using box breathing (breathe in: hold: exhale) from 1 to 10 seconds or breats, to assist with bringing my HR down after hard efforts or races and before going to sleep. It’s very calming and helps with my sleep routine.