In the opening and middlegame stage king requires protection but as soon as game reaches to last phase, king activation has significant effect over the result of chess game. For better endgame result you should activate or centralize your king as soon as possible as there are less chances of mating attack due to few materials left on the board where you can use your king as an extra piece.
Mikhail Shereshevsky has devoted a separate chapter on 'Centralization of king' in his book endgame strategy where he gave general rules to ameliorate your chess endgame.
“The great mobility of the King forms one of the chief characteristics of all endgame strategy. In the middlegame the King is a mere 'super', in the endgame on the other hand it is one of the 'principals'. We must therefore develop him, bring him nearer to the fighting line.” - Aron Nimzowitsch
An experienced chess player is mostly playing with an extra piece in endgame compare to novice. Let's go through some examples which supports this thumb rule.
Opening Tricks And Traps Part - 2