A exhaust-gas heat management system includes a catalytic converter in the exhaust-gas train of an internal-combustion engine, a cover enclosing the catalytic converter, and thereby realizing a cavity for holding a latent-heat storage PCM, at least two fluid connections between the cavity and the collecting vessel, and a pump device for activating and deactivating a PCM circuit between the cavity and the collecting vessel by means of the fluid connections. A method comprises determining an operating state of the internal combustion engine, determining the catalytic converter temperature, determining the PCM temperature, activating the PCM circuit if the PCM temperature is above a phase transition temperature of the PCM, and the internal combustion engine is in a switched-on operating state or the internal combustion engine is in a switched-off operating state and the catalytic converter temperature is below a light-off temperature of the catalytic converter.