Public awareness has an important effect on the legislation and implementation of climate change policies. Against the backdrop of the "Big Data Era," social media is an appealing and promising tool for a timely and complete understanding of public perception and attitudes towards climate policies. This paper examines the public's spontaneous attention and awareness about carbon emissions trading (ETS). Tweets related to the EU-ETS, published between 2008 and 2019, were collected for multi-dimensional analysis. Empirical results show several important findings. First, public attention on the EU-ETS has increased significantly since 2011. Second, government officials and industry practitioners have a stronger influence in the discussions than the public and industrial enterprises. Third, topic followers mostly gathered in Belgium (16.65%), the UK (11.6%), and some non-regulated countries like the US and Australia. Fourth, the public mainly focused on the policies and legislation, allowance price, and allocation. The innovation of this study rests in the development of a social media data-based research framework to examine the public's cognition of climate policies, which integrates the advantages of public social media, social network analysis, and text topic analysis. This study provides comprehensive analysis and support for climate policy implementation and public acceptance improvement.