As special as pets are to our families, in the state of Arizona, they are considered property.
The owner of the pet prior to marriage, (for the most part) can consider the pet their personal property. Alternatively, if the pet is acquired during the marriage the pet is considered marital property.
It’s not unusual for pets to travel between the new residences with the child(ren), or for the parties to share the responsibility of caring for or spending time with their pet(s).