When the Persian emperor Cyrus the Elder restored the exiles to Jerusalem, he installed Zerubbabel as governor and provided funds to rebuild on the ruins of Zadok’s temple. The high priest in charge was Jehozadak’s son, Joshua. https://pewpewhq.com/trng/the-houses-of-god

Kings were not allowed to enter the Dwelling Place of God.
It’s a little weird to call something by Solomon's name if he never entered it… It aligns more with Biblical tradition to call this first one Zadok’s Temple https://pewpewhq.com/trng/the-houses-of-god

The first dwelling place of God was the tabernacle, a portable tent that the Israelites constructed in the wilderness and carried with them wherever they went until being given a more permanent home at Shiloh, in the tribal allotment of Ephraim. https://pewpewhq.com/trng/the-houses-of-god