'Your throne, O God'
Whether Psalm 45:6 addresses the king as God or reads 'your throne is God', and whether Hebrews 1:8 applies the divine address to the Son.
Psalm 45:6-7
6Your throne, God, is forever and ever. A scepter of equity is the scepter of your kingdom.
7You have loved righteousness, and hated wickedness. Therefore God, your God, has anointed you with the oil of gladness above your fellows.
Hebrews 1:8-9
8But of the Son he says, “Your throne, O God, is forever and ever. The scepter of uprightness is the scepter of your Kingdom.
9You have loved righteousness and hated iniquity; therefore God, your God, has anointed you with the oil of gladness above your fellows.”
What each tradition says
Rabbinic / Midrashic
Rashi reads “Your throne, O God” as addressing the prince and judge: his throne shall exist forever and ever. He supports this by Exodus 7:1: “I have made you a judge over Pharaoh.”
Rashi (R. Shlomo Yitzhaki, 1040-1105), Rashi on Psalms 45:7 (Heb. numbering) — The Judaica Press complete Tanach with Rashi, translated by A. J. Rosenberg (Sefaria) — On Psalms 45:7 (Heb.; = Ps 45:6 English), comment on 'Your throne, O God'
Patristic
Augustine reads the verse as “God hath anointed Thee, O God,” so God is anointed by God; the one addressed is named God. He notes that Latin appears to repeat the same case, but Greek makes a most evident distinction between the Person addressed and the one who makes the address, saying, “God hath anointed Thee.”
Augustine of Hippo (354-430), Expositions on the Book of Psalms (condensed from the Oxford translation by A. Cleveland Coxe), in Nicene and Post-Nicene Fathers, ser. 1, vol. 8 — Psalm 45, §17 (on verse 7)
Catholic
Haydock reads the vocative "O God" (Greek O Theos), marked by the Septuagint, as showing that the Messiah is here styled God, as Hebrews 1:8 indicates. He notes that some Jews would evade this proof by saying, "God is thy throne."
George Leo Haydock, George Haydock's Catholic Bible Commentary (1859) — On Psalm 44:7 (Vulgate numbering; Ps 45:6 English)
Orthodox
No sourced position found. Checked CCEL NPNF2-09 (Hilary of Poitiers and John of Damascus — no citation of Psalm 45 anywhere in the Exposition of the Orthodox Faith), NPNF2-07 (Cyril of Jerusalem's Catechetical Lectures and Gregory Nazianzen), NPNF2-14 (The Seven Ecumenical Councils), Schaff's creeds volumes, Philaret's Longer Catechism (pravoslavieto.com) and archive.org Cyril of Alexandria vols. 1-2 — no Eastern comment on 'Thy throne, O God' (or on Hebrews 1:8's use of it) found.
Reformed / Protestant
Matthew Henry reads Psalm 45:6 as God the Father addressing the Son: “Thy throne, O God, is forever and ever.” Hebrews 1:8-9 quotes this to prove the Son is God and has a more excellent name than the angels. The Mediator must be God to do the Mediator’s work and wear the Mediator’s crown.
Matthew Henry (1662-1714), Matthew Henry's Commentary on the Whole Bible, Vol. III (Job to Song of Solomon) — On Psalm 45:6
Evangelical
Wesley says that “O God” uses God in the singular, which Scripture never uses absolutely of any but the supreme God. He says the reign whose scepter is its emblem is full of justice and equity, citing Psalm 45:6-7.
John Wesley (1703-1791), John Wesley's Notes on the Bible — Notes on Hebrews 1:8