Nowhere in scripture does it say that the anti-Christ will build the 3rd temple. In fact scripture is clear that The Lord will build the temple (Zech 6:12). If Israel has been converted after the war of Gog Magog why would it allow the beast to have such an honor? Or better yet why would God? To say anyone other than Christ will build the temple is heresy.
In Daniel 9:27 we read that the anti-Christ will confirm a seven year covenant with the many (Israel) and in the middle of the covenant will put an end to the sacrifice and offering. Both the Lord Jesus and Paul confirmed this as an end times event. Paul said He would accomplish it by standing in God’s Temple proclaiming himself to be God (2 Thes. 2:4).
In Ezekiel 44:7-8 at the outset of the Millennium,the Lord will accuse Israel of letting foreigners, uncircumcised in heart and flesh, officiate in His Temple. He will be referring to the anti-Christ and the False Prophet.
Following the Battle of Ezekiel 38-39, Israel will return to God, but will be fooled into accepting a false Messiah, the anti-Christ, as Jesus prophesied in John 5:43. I believe Israel will build a Temple, with the anti-Christ’s help, as Daniel 9:27 says. He will then make the Temple desolate, kicking off the Great Tribulation (2 Thes. 2:4 and Matt. 24:15). In a prophetic fulfillment of the Feast of Hanukkah, this Temple will be cleansed and re-dedicated at the time of the Lord’s return, fulfilling Zech. 6:12.