The thermodynamic structure of hot gas in galaxy clusters is sensitive to astrophysical processes and typically difficult to model with galaxy formation simulations. We explore the fraction of cool-core (CC) clusters in a large sample of $370$ clusters from IllustrisTNG, examining six common CC definitions. IllustrisTNG produces continuous CC criteria distributions, the extremes of which are classified as CC and non-cool-core (NCC), and the criteria are increasingly correlated for more massive clusters. At $z=0$, the CC fractions for $2$ criteria are in reasonable agreement with the observed fractions but the other $4$ CC fractions are lower than observed. This result is partly driven by systematic differences between the simulated and observed gas fraction profiles. The simulated CC fractions with redshift show tentative agreement with the observed fractions, but linear fits demonstrate that the simulated evolution is steeper than observed. The conversion of CCs to NCCs appears to begin later and act more rapidly in the simulations. Examining the fraction of CCs and NCCs defined as relaxed we find no evidence that CCs are more relaxed, suggesting that mergers are not solely responsible for disrupting CCs. A comparison of the median thermodynamic profiles defined by different CC criteria shows that the extent to which they evolve in the cluster core is dependent on the CC criteria. We conclude that the thermodynamic structure of galaxy clusters in IllustrisTNG shares many similarities with observations, but achieving better agreement most likely requires modifications of the underlying galaxy formation model.
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