Abstract: I will discuss the universal behavior of spin-resolved data of conformal field theory (CFT) at high energies. It is known that the high-temperature behavior of the grand-canonical partition function with very small angular fugacity turned on is controlled by thermal EFT on S^{d-1}\times S^1 . We will show, for d>2 CFTs, the universal high-temperature behavior of the grand-canonical partition function with finite order one (as opposed to very small) angular fugacity turned on, has a different EFT description, in terms of the thermal EFT on a background geometry with inverse temperature q\beta and spatial manifold S^{d-1}/\mathbb{Z}_q where q is the order of angular rotation. This determines the high-temperature expansion of this partition function in terms of the usual Wilson coefficients of thermal EFT, up to new subleading contributions from ``Kaluza-Klein vortices" that we classify. Consequently, we show that the effective free energy density of even-spin minus odd-spin operators is smaller by 1/2^d. This result can be considered a higher-dimensional version of a similar formula for 2D CFTs, generally derived using modular invariance. I will also discuss holographic interpretations of this result.