Rapid breathing in cats while traveling by car is a stress response. Cats will adjust to the discomfort of car travel by increasing their breathing rate. For both cats and dogs, this stress response is very unpleasant. Some animals may even vomit. 
Measures to prevent stress reactions in cats after they get in a car:
Fast your cat before it gets in the car to avoid stress reactions that could cause it to vomit during the journey.
