Climate change communication power is an important variable to characterize the influence of climate change information. To deeply understand the public's climate change communication power in China under the “dual carbon” context, this study selected Weibo as the source for obtaining climate change texts and analyzed the spatial-temporal differences of climate change communication power by using a web crawler and a communication power model. Through the comparative analysis of the two periods before and after the “dual carbon” goal was put forward, it is found that: (1) After the “dual carbon” goal was put forward, the public's discussion on climate change was more widely distributed, which formed a concerned circle of climate change in coastal areas, but the extensive discussion on climate change did not mean the enhancement of climate change communication. (2) Celebrities play an important role in the communication of climate change, but users with strong communication power for climate change do not mean that the amount of forwarding must be high. (3) The communication power of climate change is generally in a state of fluctuation, and economically developed areas have stronger communication power. (4) Keywords such as “carbon footprint” and “green technology” are the most powerful in the situation after the “dual carbon” goal was put forward. Based on the comparative analysis, the study grasps the role orientation and division of labor as a whole, which will provide a basis for formulating regional climate change-related policies more pertinently. The introduction of z-index into the study of climate change transmission in Weibo is an innovation in the application scenario of z-index.