AA member, say, I installed in cooler room (5)

I believe the answer is:

attic

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'room' is the definition.
(type of room)

I don't understand how the remainder of the clue works.

(Other definitions for attic that I've seen before include "Athenian - garret" , "Lofty room" , "Of Athens" , "of Greek region" , "Floor" .)

I've seen this clue in The Independent.

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