Breathless
At the same rider fitness session that we did the "bring sally up" squat track, the instructor was talking about breathing. The detail was in the mounted section, which I was watching and couldn't always hear....but...
One of the exercises she got the riders doing was to walk a circuit of the arena and count how many "breaths" it was. Then repeat in trot and count again.
She said that generally the count in trot is half that of the walk, and suggested that was because the trot is 2 time and the walk 4 time. But I got thinking about what impacts the number of breaths in a known distance (one circuit of an arena) - the main factor is simply going to be speed of travel, implying that generally the working trot is around twice the speed of the (working) walk.
Yes of course the rider's state of mind will impact their respiration rate. And the horse will pick up on that - whether calm and slow or tense and fast. And if a rider is more tense at higher speeds, you would see that show up in an increased number of breaths for a given distance. Or indeed if the rider starts to hold their breath e.g. over a jump. Many many years ago I did an Alexander Technique workshop and realised I had a tendency to hold my breath when I make "decisions". This isn't necessarily about big life decisions, but simple things like direction, speed, destination. The catch 22 is that holding your breath restricts the oxygen to your brain and won't assist those decision !
There's a slightly different exercise I've seen Mark Rashid do with riders, where you count how many of the horse's strides you get to each breath. (This only requires the rider to count for a shorter time - a number of the riders were losing their count on the way round the arena !) He then talks to the rider about taking a fuller slower breath, which means more "strides per breath".
The day before the rider fitness event, I was observing a Joe Midgley clinic. He talked about using his diaphragm when working the horse from the ground to cue a change in speed (up or down). While this is about more than "just" breathing.... it will be part of the overall cue.