The majority of jurisdictions in the United States do not recognize a cause of action for pain and suffering or emotional distress due to injuries to pets. In most jurisdictions animals are considered chattels and, as such, claims for their injuries are limited to the value of "repairs" or, at best, their market value. (See Texas, for example; but also see Colorado for the opposite view.)
In contrast, there is news today that in France, the General Assembly is moving to change the Civil Code to recognize that animals are living beings capable of feelings. Presumably this would allow owners to recover for injuries to their pets, but it is not clear to me whether the claims would be for injuries suffered by the pet or for injuries suffered by the owners due to injuries suffered by the pet. Go here for a discussion of the issue.