From $Z$ to $a$: High-temperature relations, subleading semi-universality, and conformal anomalies

The free energy of any CFT, $ \ln Z(β; ω_i)$, admits two expansions: high temperature ($β\rightarrow 0$) and fast rotation ($ω_i \rightarrow 1$). We demonstrate that locality of the thermal effective action forces $\ln Z$ to take a simple analytic form at all orders in the high temperature expansion