The question is more an individual one because there are no strict rules. Water fountain pumps are designed to work around the clock. There may be exceptions, it depends on the pump manufacturer. It is harder on the pump if it is constantly turning on and off. Therefore, for the equipment to last much longer, it is better to have the fountain pump running all the time.
You don’t need to turn the fountain off if it has enough water for a certain amount of time. As a rule, the manufacturer of the equipment gives recommendations on the rules of operation. It is enough just to adhere to the instructions so as not to damage the fountain.