Abstract We investigate the critical phenomena of Kerr-AdS black holes in the modified first law of thermodynamics.
The modified black hole thermodynamics exhibits the van der Waals-like phase
structure. All the critical exponents are calculated, and the swallowtail diagram of free energy is
plotted. Comparing with existing results, the main difference is the correspondence between thermal
quantities of the Kerr-AdS black holes and the van der Waals system. In previous work[16],
the correspondence was $(\Omega_H, J)\to (V, P)$. In our work, the correspondence is $(J, {\hat\Omega}_H)\to (V, P)$.
This difference results from rotating effect. The modified black hole thermodynamics is associated
with rotating observers. The free energy in such reference contains an extra rotating energy. This
extra part induces a Legendre transformation in $({\hat\Omega}_H, J)$ cross-section, which yields the different
correspondence.
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