The Court of Azimuth was an old Lombax citadel located in the Lombax Ruins on Fastoon. It was architected by Lombax General Alister Azimuth. When Emperor Percival Tachyon attacked the city after his rise to power, many Lombaxes took refuge in the Court where they decided to send themselves to another dimension in order to escape. They did this by using the Dimensionator.
Ratchet & Clank Future: Tools of Destruction
Clank received a vision from the Zoni that showed Ratchet standing in front of a swirling blue portal of energy inside the Court of Azimuth. This gave Clank reason to suspect that Tachyon was heading to Fastoon after his attack on Meridian City. His suspicion proved to be correct as Tachyon attacked Fastoon again to set more Cragmites free from their interdimensional exile.
While fighting through the Cragmite forces on Fastoon, Tachyon told Ratchet to meet him in the Court of Azimuth where he would tell Ratchet about the Lombaxes whereabouts. Once Ratchet arrived Tachyon explained how Ratchet was left behind from the other Lombaxes because of the actions of his father, Kaden. Tachyon gave Ratchet the option to join the Lombaxes in their current dimension, opening a portal that led there and making Clank's vision become a reality. Ratchet refused, knowing that stopping Tachyon was more important, and engaged in combat with the Emperor inside the Court of Azimuth. After a brief fight the Dimensionator malfunctioned and sucked Tachyon in a wormhole leading to a distant field of asteroids. However, Tachyon was far from defeated and pulled both Ratchet and Clank through to continue their battle.
After Tachyon was finally beaten Ratchet and Clank left the Court of Azimuth to head back to the Apogee Station.
It's unknown what became of the Court after the events of Tools of Destruction, although it is generally presumed that it still remains.
Ratchet & Clank Future: A Crack in Time
The Court of Azimuth appeared in a flashback, standing amongst other burning buildings as Alister Azimuth recalled the attack on Fastoon and explained his part therein all to Ratchet.