The seal runner has an annular body secured to a rotating shaft of the gas turbine engine whereas ring segments are secured to a case of the gas turbine, with the seal runner having a radially-outer surface having a contacting portion adapted to rubbingly receive ring segments of the contact seal assembly during use, the seal runner having a radially-inner surface opposite to the radially-outer surface. The method includes rotating the seal runner relative to the ring segments and generating heat from the rubbing engagement therebetween; and feeding a flow of cooling fluid against the radially-inner surface to cool the seal runner from said generated heat including maintaining a pool of cooling fluid having a given depth against the radially-inner surface.