Why Solomon's temple? What does it have to do with the Christian faith?
The temple that Solomon built was magnificent, and the glory of the Lord filled it 2 Chr 7:1 But it was only good because of the LORD. Otherwise it was just big stones and gold plating, valuable sure, but also meaningless, as Solomon himself would attest in Ecclesiastes.
Under Roman occupation, around the time that Jesus was born the Herod the Great was reconstructing the temple. Perhaps for reasons similar to those outlined by Dick Hartfield in another answer: to gain legitimacy. When one of Jesus' disciples noted how magnificent the temple was Jesus answered him: 'Do you see these great buildings? There will not be here one stone upon another that will not be thrown down'. In that passage Jesus foretold of his return and linked it to the even in 70 AD when the Romans destroyed the temple.
Jesus' prediction of the destruction of the temple was also one of the charges brought against him, but actually, he was using the word 'temple' to refer to his own body.
In Christianity we see the temple as a type of the real temple of God, which is Jesus, and indeed any human body in which the Spirit of God dwells. Having an ostentatious building might even be counterproductive in the Christian faith since it focusses our attention on this world, instead of looking toward the invisible God.
I was more interested in a deeper explanation of why they are choosing elements of the Old Testament.
There is a lot in the Old Testament that we ought to attend to: the repeated commands to justice, mercy and righteousness, the commands to fear the LORD, love one's neighbour and so on. But there are some elements that it would be wrong for a Christian to indulge in: cirumcision and animal sacrifice. Given Jesus' expressed disdain for the great temple building project in his day it might be appropriate to add temple building to the list.
And to indulge in a bit of end-times speculation. The temple builder might be trying to make some Jewish folk a bit jealous so that they hasten to build the third temple in Israel and restore the sacrifices until the Antichrist installs himself in the temple, claims that he is God and finally we can get on with the schedule outlined in scripture that culminates in the return of Christ Jesus!
EDIT: 11 Mar 16
Just to add another perspective to the end times speculation, some say that there is a wide misunderstanding of prophecy and the antichrist referred to in scripture, is actually a system of false belief that has been around for a long time. On this reading there is no need to look for a rebuilding of Solomon's temple near the end, for the sacrifice has already been abolished when Jesus performed the sacrifice of his sinless self.