Revisiting particle circular orbits as probes of black hole thermodynamics

Recent studies propose that black hole phase transitions can be encoded in the circular orbit radius of particles. In this paper, we systematically investigate the reliability of this encoding mechanism. We find that this mechanism is highly reliable in the isobaric ensemble, whereas it may break down in the isothermal ensemble. It turns out that the reliability of this mechanism is directly controlled by the first law of black hole thermodynamics. Interestingly, even if this encoding mechanism fails, we prove that, for any black hole exhibiting criticality, the first law can ensure that, near the critical point, the coexistence gap of the circular orbit radius remains a reliable order parameter and yields exactly the same critical exponents as the standard thermodynamic order parameter. Our results provide a potential way to identify the thermodynamic ensemble of a black hole, and reveal a deeper connection between gravitational geometry and thermodynamics.