Based on the Prisoner's Dilemma game, we study the effect of the diverse strategy-updating time scale on the evolution of cooperation under the normalized payoff framework. Agents can adjust their strategy-updating speed according to their fitness and collective influence, and this mechanism promotes the emergence of cooperation on Barabasi-Albert scale-free networks. Moreover, agents with higher values of collective influence may have longer persistence-cooperation duration and diffuse their cooperative behaviors among neighbors efficiently. Through investigating the game-learning skeleton, we find that the heavy-tailed in-degree distribution emerges and influencers with proper depth length play an important role in maintaining cooperation.