A device for cooling a turbine casing in a turbomachine including a turbine is disclosed. The turbine includes several stages, at least one of the stages includes a nozzle assembly formed of an annular row of fixed vanes and an impeller mounted to rotate in a cylindrical shroud formed of ring sectors fixed to the casing, a cooling circuit including ducts carrying cooling air into cavities formed in the vanes of the nozzle assembly, and an air-carrying arrangement which carries air to casing upstream hooks for suspending the ring sectors.