The Irishman, 34, has been drafted in by the Cumbrians to cover for injuries to both Chris Lumsdon and Paul Thirlwell.
Manager John Ward told the club's official website: "Graham is a very good player and he also comes with experience and knowledge.
"He can help us on the pitch with the way that he can play the game and he can also bring that experience into it as well.
"He's actually one of those players that you would sign without seeing him, because he brings so much into a team.
"He gives us cover for our injured players, of course and I think he has the type of character that will help to give everybody a lift."
Kavanagh has not featured for the Black Cats this season, having finished last term out on loan at Sheffield Wednesday.