Keep in mind...the soil cools off slower than the air. The roots are still assimilating nutrients. Photosynthesis goes on even in cold weather. So the grass gets dark green, roots grow deeper and get fatter as starch is accumulated and stored in the roots for next year. A certain amount of nitrogen is probably stored also.
Our local professors suggest slow-release nitrogen. Quick release is fine--BUT--heavy rains can result in leaching of the quick-release nitrogen deep into the soil--eventually contaminating ground water.
I wouldn't add phosphorus, nor much potassium unless a soil test indicated a need. The idea that either will stimulate roots is not supported by most turf professors.