The temple building does not need to exist for an "abomination of desolation" to be an "abomination of desolation", because Mount Moriah itself is set-apart ground, because that is where the temple building once stood that set that ground apart. That is why the abomination of desolation that was appointed to come after the 1,290 years of Daniel came not into fulfillment with a standing temple, but by the erection of the Dome of the Rock where the temple once stood, which is Mount Moriah, for the 1,290 year timeline began around 600 BC, which is one instance temple sacrifices stopped, which was around the beginning of the Babylonian captivity, to around 691 AD when the construction of the Dome of the Rock was completed, thus 1,290 years,
Jehoiakim was twenty-five years old when he became king, and he reigned in Jerusalem eleven years. He did evil in the eyes of Yehovah his God. Nebuchadnezzar king of Babylon attacked him and bound him with bronze shackles to take him to Babylon. Nebuchadnezzar also took to Babylon articles from the temple of Yehovah and put them in his temple there.
2 Chronicles 36:5-7
From the time that the daily sacrifice is abolished and the abomination that causes desolation is set up, there will be 1,290 days.
Daniel 12:11
And so, just as that abomination of desolation of the 1,290 year timeline came not with a standing temple, so in likewise manner the abomination of desolation of the Romans also may not have come into fulfillment with a standing temple either, although I suspect it did, because the details of what actually happened may have been lost with time, but we do have credible evidences that confirm that an abomination of desolation did occur by the hand of the Romans on Mount Moriah, for the eastern gate of the temple was indeed Mount Moriah, for that was the entrance to the temple itself where the altar and washing basin were, all in fulfillment of prophecy.