“the house.” This is the Temple in Jerusalem.

“and stopped over the cherubim.” This is a very brief and clipped description, but clear enough in the context of Ezekiel’s visions. God had parked His cherubim chariot-throne in the courtyard of the Temple, south of the Temple itself (Ezek. 10:3). Above the cherubim was the sapphire-colored platform, and on the platform was God’s throne. So this short phrase is describing Yahweh moving from the threshold of the Temple where He had been (Ezek. 9:3; 10:4), and getting on His throne, which is on the platform over the cherubim, and in the next verse Yahweh, on his chariot-throne, moves on to the next location, which we learn in the next verse, Ezekiel 10:19, is the entrance of the east gate of the Temple. This “east gate” is the far east gate on the eastern wall of the Temple complex. So the sin of the leaders and people of Judah drove Yahweh from His place between the cherubim in the Holy of Holies to the threshold of the entrance of the Temple, and then to the east gate of the Temple. God is steadily leaving His Temple. God’s next stop as He leaves Jerusalem, and the last one recorded in Ezekiel, is over the Mount of Olives east of Jerusalem (Ezek. 11:23). There is no record of the glory of God returning to the Temple built by the people who returned from the Babylonian Captivity as recorded in Ezra, the Second Temple that was embellished by Herod the Great and destroyed by the Romans in 70 AD. The glory of God will return to the Millennial Temple built by Jesus Christ as per the prophecy of Ezekiel 43:1-4.

[For more on Ezekiel 8-11 and what happened, see commentary on Ezek. 8:4.]