Why would anyone want to interpret the Greek words "ho theos" as "O God" in verse 8 but the same words as "the God" (2x) in verse 9?
God's throne represents God's authority. David/Solomon sat on God's throne executing God's authority over Israel (1 Chronicles 29:23). Hebrews 1:8 refers to the fact that Jesus sat down on God's throne which means he executes God's authority in God's name, "Your throne ho theos to the age of the age."
The Trinitarian translation results in absurd implications. It results in God's God anointing God so that God could make God above God's peers. It also results in the highly unlikely result of calling an ancient Davidic king "God," a king who was contemporaneous with the Psalmist. There are several passages like Psalm 45 in Scripture which refer to both a contemporary figure and to Jesus.
Examples: 2 Samuel 7:11-14 - both Solomon and Jesus Isaiah 7:14 - child born during the reign of Ahaz, and Jesus Isaiah 42:1ff - both Israel and Jesus Isaiah 53 - both Israel and Jesus (Isaiah personifies Israel as a single man many times in the book of Isaiah) Hosea 11:1 - both Israel and Jesus