First, the Dome of Rock is not a Mosque. It is a shrine. It was built in the late 7th century, shortly after the Arabs took the Holy Land from the Byzantine empire. At the time, the Byzantines were using the Temple Mount as a garbage dump. The Arabs rightly found this appauling and forced the local rulers to clean it up. They then built the Dome out of respect for the site, which they revered because it was the site of the ancient temple, but they did not yet come to believe it to be the spot from which Mohammed ascended to heaven. It took another century or so for that belief to develop.
My second point is addressed to my fellow Christians who support Israel demolishing the Dome of Rock and renewing the Levitical sacrifices. Have you, brother and sisters, ever read the book of Hebrews? The sacrifice of Our Lord on the Cross suppercedes all Old Testament sacrifices and makes them useless. This is not only explicitly clear in Hebrews, but it is also the universal teaching of all the Church Fathers since time immemorial. Therefore, I believe your desire to see the temple rebuilt is quite misplaced, and perhaps based on a misreading of some of the apocalyptic sections of the Bible.
Misplaced indeed. Good message..