My pregnant dog is breathing fast and digging. Is this early labor?

It sounds like she might be showing signs of early labor, but it also could be normal behavior for the end of pregnancy. I would recommend seeing your veterinarian for an exam and an x-ray if you haven't already seen them recently - it is very helpful to take an x-ray to count puppies late in pregnancy. That way you know approximately how many puppies to expect. You should also start checking her rectal temperature with a well lubricated digital thermometer three times a day to establish a baseline (make sure it's at the same time every day) - usually a dog's baseline temperature will drop by one full degree prior to labor. If Luna's temperature drops but you don't see puppies within 24 hours you should see a veterinarian to make sure everything is okay.