Will the Third Temple be built?

Perhaps.

From Texas to Israel: Red heifers needed for Temple arrive:

the Temple Institute continues preparations to lay the ground for the construction of the Third Temple in Jerusalem.

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But for me, the Third Temple is not needed, Hebrews 10:

1 For since the law has but a shadow of the good things to come instead of the true form of these realities, it can never, by the same sacrifices that are continually offered every year, make perfect those who draw near. 2Otherwise, would they not have ceased to be offered, since the worshipers, having once been cleansed, would no longer have any consciousness of sins? 3 But in these sacrifices there is a reminder of sins every year. 4 For it is impossible for the blood of bulls and goats to take away sins.

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The blood of red heifers cannot take away sins permanently. But there is good news:

5 Consequently, when Christ came into the world, he said,
“Sacrifices and offerings you have not desired,
but a body have you prepared for me;
6 in burnt offerings and sin offerings
you have taken no pleasure.
7 Then I said, ‘Behold, I have come to do your will, O God, as it is written of me in the scroll of the book.’” ...
11 And every priest stands daily at his service, offering repeatedly the same sacrifices, which can never take away sins. 12 But when Christ had offered for all time a single sacrifice for sins,

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on the Cross

he sat down at the right hand of God, 13waiting from that time until his enemies should be made a footstool for his feet. 14 For by a single offering he has perfected for all time those who are being sanctified.

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Jesus' cross did away with the need for the Temple.

18 Where there is forgiveness of these, there is no longer any offering for sin.

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Will the Third Temple be built?

I don't know. Right now, the Islam Dome of the Rock is on the site of the Jewish Second Temple. It would be crazy news if the jews started building the Third Temple. However, I would not give a penny to support the project. I don't need the temple, and neither do any Christians.

See also

• Will Ezekiel's temple be built?
• My HOUSE will be called a house of prayer for ALL the nations