A. I will reiterate what Dr. Edwards answered below. Mouth-breathing is abnormal in cats, and usually indicates respiratory (lung) or heart disease. Even when cats have an upper respiratory problem, such as a viral infection in the nose and sinuses, they do not breath out of their mouths. Possibly causes include feline asthma and heart disease. Although listening to the lungs and heart with a stethoscope can give us clues that there is a problem, ultimately problems are only obvious on x-rays. If your cat has not had chest x-rays, that is surely the next step.
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