Accurate identification of effective epidemic threshold is essential for understanding epidemic dynamics on complex networks. In this paper, we systematically study how the recovery rate affects the susceptible-infected-removed spreading dynamics on complex networks, where synchronous and asynchronous updating processes are taken into account. We derive the theoretical effective epidemic threshold and final outbreak size based on the edge-based compartmental theory. To validate the proposed theoretical predictions, extensive numerical experiments are implemented by using asynchronous and synchronous updating methods. When asynchronous updating method is used in simulations, recovery rate does not affect the final state of spreading dynamics. But with synchronous updating, we find that the effective epidemic threshold decreases with recovery rate, and final outbreak size increases with recovery rate. A good agreement between the theoretical predictions and the numerical results are observed on both synthetic and real-world networks. Our results extend the existing theoretical studies and help us to understand the phase transition with arbitrary recovery rate.