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Breathing heavy is usually referred to as panting. Many different things can cause constant panting. Heavy panting can be caused by heart problems, lung problems, fever, or bronchitis. You can read about this here:
Dogs pant for various reasons. They pant to cool down, but they also pant when they are under stress, in pain, or afraid. There can also be medical causes such as neurological problems, respiratory disorders, and Heart problems such as heartworm, anemia, and fever.
If your dog is panting but has no other symptoms, keep a watch on him. If it still seems excessive after a day or so, or if he develops more symptoms, I'd schedule an appointment with your Vet.. Given that it is happening when he is asleep, It would not be due to being afraid or under stress, so I would mention it to your vet and be sure his heart and lungs are checked.
I hope this information is helpful to you.