A cool box is a fridge only if it holds at or below 5°C, and almost nobody checks. Ice in the box feels like refrigeration and frequently is not – by the afternoon a lot of cool boxes sit in the teens, which is inside the range where bacteria grow rather than below it.
A few pounds of thermometer turns “probably fine” into a number. If it holds, normal food safety guidance applies and you can plan around leftovers. If it does not, you do not have a fridge and should cook to the meal instead. Either answer is worth knowing; the guess is not.