Your pet lives for 15 hours in the game. You can extend this time limit by going into a camp that you make, then the Pokegatchi Pen. Afterwards, select "Set Pet Free". Your pet will appear on screen. Re-capture it, but you must clean it up as it will be just as hungry, dirty, etc. as when you first caught it. The time will then go back to 0:00 for the age.