One of the critical emerging challenges in climate change communication is the prevalence of conspiracy theories. This paper discusses some of the major conspiracy theories related to climate change found in a large Twitter corpus. We use a state-of-the-art stance detection method to find whether conspiracy theories are more popular among Disbelievers or Believers of climate change. We then analyze which conspiracy theory is more popular than the others and how popularity changes with climate change belief. We find that Disbelievers of climate change are overwhelmingly responsible for sharing messages with conspiracy theory-related keywords, and not all conspiracy theories are equally shared. Lastly, we discuss the implications of our findings for climate change communication.