I might think that also included are the Temple offices, the Temple courts, and the Temple mount. It is, therefore written "house." Just as "house" is mundane (and not sacrosanct), so all that are mundane (are included in the mezzuzah requirement).
Beloved are Israel, whom the Holy One Blessed be He surrounded with (seven) mitzvoth: Tefillin on the head and tefillin on the arm, mezzuzah on their doors, and tzitzith on the four corners of their garments. About them David said (Psalms 119:114) "(For these) seven in the day have I praised you." He entered the bathhouse and saw himself naked — whereupon he said: "Woe unto me that I am 'naked' of mitzvoth," but looking upon his circumcision, he uttered praise, viz. (Ibid. 12:1) "For the chief musician on the eighth, (circumcision, the eighth mitzvah, on the eighth day), a psalm of David."