Simon J. Kistemaker writes that Amos 9:11, quoted in Acts 15:16, "is a prophetic reference to the temple of the Lord."
Simon J. Kistemaker, Exposition of the Acts of the Apostles (New Testament Commentary 17; Grand Rapids, MI: Baker Books, 1990), 554 (Logos ed.)
The phrase the tent of David is a prophetic reference to the temple of the Lord to which all the nations go to worship God (compare, e.g., Isa. 2:2–4; Zech. 14:16). The prophets predict that both Jews and Gentiles together worship God. Note that Amos links the word tent to David, not to Levi or Aaron. David is a witness to the peoples on this earth, so that nations who do not know God will run to him (Isa. 55:3–5). These prophecies, which David’s descendant, Jesus Christ, eventually fulfills, are messianic. By mentioning David’s tent, Amos sees the panorama of Gentile nations coming to know and worship God. At the time of the Jerusalem Council, James indicates that this messianic prophecy of Amos has been fulfilled with the entrance of Gentiles into the church. James teaches that Israel, restored through Jesus Christ, extends a welcome to the rest of mankind in spiritual fellowship