Recent instances of political backlash against global governance efforts as well as conventional wisdom suggest that there is a link between shifting authority from the domestic to the global level, on the one hand, and the legitimacy of global governance institutions as perceived by citizens and other stakeholders on the other. We thus investigate whether and how increasing the authority of a global governance institution affects citizens' legitimacy perceptions, using a population-based survey experiment in Germany and the United States (N=1600 each). The empirical focus is on climate change, a costly and paradigmatic global governance effort. The results show that certain shifts of political authority, such as changes to majority decision making at the international level and automatic implementation of international decisions domestically, do not significantly affect "average" citizens' legitimacy perceptions of global governance institutions. This result is not due to citizens' incapacity to understand the implications of increasing authority, namely, that increasing authority results in a loss of control over climate policy in Germany and the United States. Rather, legitimacy perceptions appear to be shaped by citizens' perceptions of procedural and performance quality of such efforts as well as by their level of cognitive mobilization, namely their interest in international politics. In brief, we find that citizens relate perceived procedural and performance quality of global governance with their evaluation of its legitimacy, but that subtle shifts of authority from the domestic to the global level do not per se affect citizens' legitimacy perceptions.