Gave them up (paredōken). First aorist active indicative of paradidōmi. This same form occurs three times like clods on a coffin in a grave in Rom 1:24, Rom 1:26, Rom 1:28 where Paul speaks of God giving the heathen up to their lusts.
To serve the host of heaven (latreuein tēi stratiāi tou ouranou). The verb latreuō is used of the worship of God (Mat 4:10) as well as of idols as here (from latron, hire, latris, hireling, then to serve). But the worship of the host of heaven (Deu 17:3; 2Ki 17:16; 2Ki 21:3; 2Ch 33:3, 2Ch 33:5; Jer 8:2; Jer 19:13) is Sabaism or worship of the host (stratia) of heaven (sun, moon, and stars) instead of the Lord of hosts. This star-worship greatly injured the Jews.
In the book of the prophets (en biblōi tōn prophētōn). That is the twelve minor prophets which the Jews counted as one book (cf. Act 13:40). This quotation is from Amo 5:25-27. The greater prophets were Isaiah, Jeremiah, Ezekiel.
Slain beasts (sphagia). Here only in the N.T. (from Amo 5:25) sphagē, slaughter, sphazō, to slay.