b.Berachot 3a makes the connection between Hashem’s “roaring like a lion” during the three watches of the night, to the destruction of the Beit Hamikdash. It is said that He cries out at each of the watches of the night, “Woe to the children because of whose sins I destroyed My Temple, and burned My Sanctuary, and exiled them among the nations of the world.”
From this it is derived that there are three “favorable” times of the night at which one is to pray, particularly for teshuvah and for the rebuilding of the Beit Hamikdash: The first watch (soon after dark), the second watch (the middle of the night), and the third watch (just before dawn). I believe this is where the Breslovers get the concept of waking up to pray at midnight, and it may be connected to the Jewish legend that David HaMelech arose at midnight to pray (being awakened by the sound of his harp being played by the רוח).
Toras Chaim (to Sanhedrin 96b) states that the expressions, “destroyed My Temple” and “burned My Sanctuary” are representative of two events: Hashem’s departure from His Temple, and the physical destruction of the Temple, respectively.
There is also a thought that not only was the physical Temple destroyed, but that the heavenly Temple ceased services while the earthly Temple remains in ruins (Taanit 5a).
I’m definitely interested in hearing from someone if they have any insights on these.