The emerging 360-degree video has a good practical prospect in video broadcasting due to its more immersive experience than traditional videos. To maintain a high fidelity with panoramic vision, 360-degree video adopts a significantly increased resolution than planar videos, which also inevitably leads to higher encoding complexity. In this paper, we propose a flexible complexity optimization method for 360-degree video coding, in which the encoder is capable of adapting to different complexity constraints in diversified broadcasting scenarios. In particular, the proposed method is realized with a latitude-based computation complexity allocation. Firstly, it investigates the Coding Unit (CU) partition variation along the latitude (i.e., vertical axis) and divides each video frame into Largest CU (LCU)-based latitude regions. Then, it formulates the complexity allocation problem from the global complexity target to all latitude regions, by considering the intraframe dependencies and Rate-Distortion (RD) models. Finally, it calculates the solution to this complexity allocation problem and further employs it in 360-degree video coding. Comprehensive experiments on the standard dataset reveal the superiority of our method, which achieves a flexible computational time reduction from 14.53% to 68.36% whilst maintaining a high visual quality of compressed videos.