Cathedral of the Holy Cross, Acre
The Crusader period began for Acre in 1104, some five years after the fall of Jerusalem. Baldwin I, King of Jerusalem, and the Genoese commercial fleet cooperated in laying siege to Acre from land and sea until the city fell to the Crusaders. […] The military orders installed themselves nearby - the Hospitallers in the north of the city to the west of the palace, the Templars in the south-west of the city, and the German Knights close to the eastern wall of the wall. Other important quarters included those of the Patriarch (in which the Church of the Holy Cross, Acre’s main cathedral, was built), the French merchants (the Provençal quarter), and the English merchants. Moslem and Jewish merchants also settled in the city. [x]
In 1191, with the reconquest of Acre by the Crusaders, the cult of the True Cross was reestabilished in Acre. Not only was the patriarchal church in Acre dedicated as the Cathedral of the Holy Cross, but also the headquarters of the knights Hospitaller was relocated there. […] At the time of its reconquest, Acre became the provisional capital of the Latin Kingdom and, although the Crusaders could not have known it in July 1191, it would remain the facto capital for one hundred years. […] Richard had been given two relics of the True Cross while the Crusader army was camped in Beit Nuba in June of 1192. Presumably one or both of these relics where deposited in the Cathedral of the Holy Cross before Richard left Acre.
"Crusader Art in the Holy Land, From the Third Crusade to the Fall of Acre", by Jaroslav Folda [x]
[Another difficult reserach. Since in 1291 the city was completely destroyed, nothing remains of the Cathedral today. This is all I found about it.]