AA member, say, I installed in cooler room (5)
I believe the answer is:
attic
'room' is the definition.
(type of room)
I don't understand how the rest of the clue works.
Can you help me to learn more?
(Other definitions for attic that I've seen before include "Athenian - garret" , "Lofty room" , "Of Athens" , "of Greek region" , "Floor" .)