A high-pressure gas discharge lamp with a cooling arrangement is described, which is characterized in particular in that the lamp can be operated at an increased power, an increase in the temperature of the coldest spot in the lamp interior generating a higher gas pressure, while the cooling arrangement (7, 71, 83, 82) is constructed and dimensioned such that a devitrification of the lamp bulb and a condensation of the filling gas are substantially prevented at said increased power. A lighting unit with such a high-pressure gas discharge lamp is further described, as is a power supply unit for operating the lamp. This not only considerably improves the spectral properties of the light, but the lamp also operates at a higher operating voltage because of the higher gas pressure, so that a correspondingly higher lamp power is achieved for a given lamp current. On the other hand, given the same lamp power, a weaker current is required, so that the electrodes will have a substantially longer useful life. All this is achieved without any change in the geometry of the lamp.