I have an approximately 4 year old Luxaire furnce that heats the house fine at times, but sometimes once the thermostat kicks to on the furnace starts up and then shuts down within a few minutes. After that the house gets cold because the thermostat thinks the furnace is on, but it's not.
Are you sure the t-stat is shutting down the furnace? It sounds like your flame sensor is dirty. Do the burners light and then drop out after about 3 sec.? You need to watch the cycle of operation and report back.