An electrochemical apparatus and method for cooling the same. The apparatus comprises a stack of at least two compact cells. The cells have a maximum thermal pathway of about 4 centimeters. The cell includes a solid electrode layer disposed between an oxygen electrode layer and a fuel electrode layer. A separator layer contacts the surface of opposing electrodes of adjacent cells. The cell defines internal passages, for providing reactant gases to the electrodes, and a rim portion of the cell is adapted to radiate heat generated within the cell to outside the cell. At least one layer of the cell is adapted to conduct the cell heat to the rim for transfer by radiative cooling. The method of cooling the above apparatus includes feeding oxygen-bearing gas to the oxygen electrode at a low flow rate, conducting the heat produced within the cell to the cell rim, and radiating the heat from the cell rim.