Extreme heat is rising globally and unequally, but evidence of effects on human cognition, social behaviour and collective well-being is still missing, and those studies that exist mostly rely on WEIRD samples and employ laboratory interventions, limiting policy relevance. We present the Heat & Cognition Project as a case study in how applied psychology can use Big Team Science (BTS) to generate scalable, context-sensitive evidence for the impact of climate change hazards. In over 40+ cities across the globe, we are implementing a preregistered three-wave protocol to test the effect of extreme heat on the human mind and collective well-being in a natural experiment, aligned to verified heatwaves and matched temperate periods (>50,000 participants). Outcome measures cover attention and working memory, mood, prosociality and social functioning, well-being, and policy attitudes, alongside real-world vulnerability and adaptation contexts of individuals. In this talk, we present first evidence from pilot data collection cities and discuss key methodological challenges when conducting global science projects related to climate science. These include defining "extreme heat" across climates (absolute vs. relative thresholds), coordinating timing by climate rather than calendar, multilingual instrument validity, and fair resource distribution. We report feasibility metrics from piloting and early waves (recruitment during short heat windows, adherence, missing-data patterns tied to access/adaptation), and preliminary cross-site patterns linking heat exposure to outcomes. We discuss how BTS can enable applied psychology to test theory under dynamic, real-world climate change stressors at policy-relevant scales.