Spin-refined partition functions and \( \mathcal{CRT} \) black holes

We investigate spin-refined partition functions in AdS/CFT using Euclidean gravitational path integrals. We construct phase diagrams for Z_X = Tr(e^−βHX) in various dimensions and for different choices of discrete isometry X, discovering rich structures at finite temperature. When X is a reflection, Z_X counts the difference between the number of even- and odd-spin microstates. The high-temperature regime is universally dominated by $\mathcal{CRT}$ \( \mathcal{CRT} \) -twisted black holes in any dimension, and in odd spacetime dimensions we examine whether complex rotating black hole solutions can contribute to spin-refined observables or potentially dominate at finite temperature. We also analyze the microcanonical ensemble. There the leading contribution almost always comes from rotating black holes, showing that the two ensembles are not necessarily equivalent.