Temperature distributions of gas in halos of TNG300, from 10⁸ to 10¹⁵ M_⊙ in mass bins of width 1 dex. The dotted lines show the average virial temperature in the corresponding mass bin. The shaded regions show the temperature regimes in the same colors as Figure 1. Even for the very massive systems that are the focus of this paper (clusters with > 10¹⁴ M_⊙, brown curve), a non negligible amount of cool gas resides within the ICM and cluster galaxies.