Consider an extreme case of high notes and low notes. A tuba player playing a loud, low forte in their range can expel between 140 - 160 liters of air per minute to generate this note. A trumpet player playing a loud, low forte in their range will expel between 6-8 liters of air per minute to generate this note. This flow rate has to do with the physical nature of brass instruments. If the tuba player jumps several octaves and plays the same note as the Trumpet player with a similar dynamic, the flow rate requirement will be between 6-8 liters of air per minute. All of this information is based on a study that Arnold Jacobs performed in a lab with brass players from the CSO back in the 1950s and is found in Brian Frederiksen's book.
If you can follow that logic, the same holds true for the trumpet as you ascend into the higher register. There is less flow rate required to generate the notes. In fact if you try to generate the same flow rate in the upper register that you do in the lower register, you will be fighting the physics of the pipe and the back pressure will be nearly impossible to overcome. The point is, you don't have to flow so much volume of air in that register. It's counter-productive!
Now, knowing this flow rate relationship to be true, think about your breathing when you are playing. If you have a long phrase in the low register, you will be emptying your lungs faster. Let’s say you are playing a phrase that requires 6-8 liters of air per minute in the low register. Most people have a vital capacity in their lungs of between 3-5 liters, so after 30 seconds of playing in the low register at a forte dynamic you will have empty lungs. If you play a similar 30-second phrase an octave and a half higher, the flow rate might only require that you expel 1 liter of air per minute. In this case your body is ready for another breath because you are at the point where the CO2 in your blood stream has risen to the point where your body says, "BREATHE!". But your lungs still have several liters of air left in them!
Learning how to deal with this is a combination of understanding what's going on (get Brian's book!) and practicing how to deal with these situations when you are playing in the different registers.