In this talk I will discuss supersymmetric Cardy formulae in d=4 and d=6. These formulae govern the universal behavior in the high-temperature regime of supersymmetric partition functions — or, in the case of the superconformal index, they govern the high-energy asymptotics of SUSY operators at large energy. I will outline the proof of the Cardy formulae for theories with moduli spaces of vacua, which relies on an effective supersymmetric Chern-Simons action in d-1 dimensions. I will argue that this effective action is universal and intimately related to perturbative as well as global gravitational anomalies. Finally, I will discuss some immediate consequences of our results and briefly compare and distinguish our results to other proposed Cardy formulas.