A mechanical seal face has a significant increase in the ability of the seal to tolerate poor lubrication conditions such as dry running. The mechanical seal employs a thin, non-corrugated flat sheet of flexible thermally conductive graphite material placed between one of the mechanical seal faces and a metal surface such as a metal surface defined by a shaft sleeve or gland serving as a seal ring holder. The graphite sheet is located axially between a back face of the more thermally conductive mechanical seal ring and the metal face holder part. The conductive seal face material preferably is a ceramic or cermet material such as silicon carbide, tungsten carbide, silicon nitride, aluminum oxide, or a metallic material such as stainless steel. Fluid pressure and spring forces are used to create a compressive load between the seal face, graphite sheet, and metal part to maximize continuous contact between the opposed faces of these parts and the opposite sides of the graphite sheet, and heat buildup at contacting seal faces flows away therefrom to the metal part through the thermally conductive sheet.