The purpose of the room is internal, and so benefit from shade cast externally shouldn't be a problem of Illegal Use of Sanctified Property: Pesachim 26a
On the 17th of Tammuz an idol was set up [by the Jews] in this room: Taanit 26b
The arrangement of an idol, by Jews, in this room, was a major cause for the downfall of the Jews: Gittin 7a