Is that what you mean?
Or do you have the idea that the Muslims somehow desecrated Temple Mount and the Hebrew Temple when they built a couple of mosques?
If so, remember that the Romans totally destroyed the whole of the Temple. The Temple Mount was yet another structure provided by Herod to support the Temple known in a lot of literature as "Herod's Temple".
After the Romans cleared the site it remained vacant and open to the elements for a long time ~ some attempts were made over the years to fix up the place, and even in the three years of Simon Bar Kokhba's rule, work was done. Of course, right after that the Romans completely razed the city of Jerusalem to the ground and built a Roman town on top of it.
The Eastern Romans did some improvements to Jerusalem and built several churches (which are still thee). Then the Dark Ages came, construction stopped, and Jerusalem dwindled to a small town (if that). Eventually it drew the attention of the Islamic Caliphate. Temple Mount was cleared of debris and two mosques were built, one of them on top of the stone that stood immediately under the Altar, and where Muslim tradition holds that Mohammad's horse leaped, with him, into Heaven.
So, how did Islam desecrate the site? Seems to me that shame belongs strictly to the Romans.