Question

) in a hot day in the summer, a "clever" thermal engineering student claims that he can keep his apartment cooler by keeping the door of the refrigerator open. could this be true and why?

Answer 1

It's not true.

A refrigerator keep the cool temperature by moving the heat from inside the box to outside the box.
If that student open the door, the temperature inside the refrigerator would be equal with the average temperature of the room, which make his effort pretty much pointless.