The thermal behavior of a thermophotovoltaic system composed of a metallodielectric spectrally selective radiator at high temperature and a GaSb photovoltaic cell in the far field is investigated. Using a coupled radiative, electrical and thermal model, we highlight that, without a large conductive-convective heat transfer coefficient applied to the cell, the rise in temperature of the photovoltaic cell induces dramatic efficiency losses. We then investigate solutions to mitigate thermal effects, such as radiative cooling or the decrease of the emissivity or the temperature of the radiator. Without extending the radiating area beyond that of the cell, gains by radiative cooling are marginal. However, for a given radiator temperature, decreasing its emissivity is beneficial to conversion efficiency and, in cases of limited conductive-convective cooling capacities, even leads to larger electrical power outputs. More importantly, for a realistic radiator structure made of tungsten and hafnium oxide, larger conversion efficiencies are reached with smaller radiator temperatures because thermal losses and thus needs for cooling are less.