Breathing patterns definitely alter according to your body temperature yes.
Although I am not certain, this may be due more to the fact that when you exercise, for example, you effectively burn more fuel in your muscles and this produces heat when free energy is released (although much more controlled by the enzymatic processes of respiration in your body, than that occurring when you burn fuel on a fire).
However, the increase in breathing that accompanies this energy expenditure during exercise is due to the demand for oxygen by the exercising tissue, not so much the increase in body temperature. Your body dissipates this excess heat through increasing peripheral blood flow (through opening additional vessels beneath the skin surface) and sweating to cool this blood though evaporation.
Furthermore, a recent paper (linked in reference on this website http://www.abc.net.au/science/articles/2008/12/16/2447520.htm) suggests that yawning may could possibly serve to cool an overheated brain, and used evidence that in very hot or very cold climates, the extremes of temperature would be detrimental to the brain if yawning occurred and so does not happen, but at temperatures closer to ambient room yawning does occur in overheated animals. Or so it seams.
However, yawning might possibly be linked to a compensatory response to changes in arterial CO2 levels resulting from variation in breathing patterns when you are tired. This has no supporting evidence, but it is possible that breathing may change in relation to temperature for a variety of reasons, which need to be tested t properly answer your question!
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