We analyzed the physical properties of the gaseous intracluster medium (ICM) at the center of massive galaxy clusters with TNG-Cluster, a new cosmological magnetohydrodynamical simulation. Our sample contains 352 simulated clusters spanning a halo mass range of 10(14) < M-500c/M-circle dot < 2 x 10(15) at z = 0. We focused on the proposed classification of clusters into cool-core (CC) and non-cool-core (NCC) populations, the z = 0 distribution of cluster central ICM properties, and the redshift evolution of the CC cluster population. We analyzed the resolved structure and radial profiles of entropy, temperature, electron number density, and pressure. To distinguish between CC and NCC clusters, we considered several criteria: central cooling time, central entropy, central density, X-ray concentration parameter, and density profile slope. According to TNG-Cluster and with no a priori cluster selection, the distributions of these properties are unimodal, whereby CCs and NCCs represent the two extremes. Across the entire TNG-Cluster sample at z = 0 and based on the central cooling time, the strong CC fraction is f(SCC) = 24%, compared to f(WCC) = 60% and f(NCC) = 16% for weak and NCCs, respectively. However, the fraction of CCs depends strongly on both halo mass and redshift, although the magnitude and even direction of the trends vary with definition. The abundant statistics of simulated high-mass clusters in TNG-Cluster enabled us to match observational samples and make a comparison with data. The CC fractions from z = 0 to z = 2 are in broad agreement with observations, as are the radial profiles of thermodynamical quantities, globally as well as when divided as CC versus NCC halos. TNG-Cluster can therefore be used as a laboratory to study the evolution and transformations of cluster cores due to mergers, AGN feedback, and other physical processes.
