effect of radiative cooling that converts part of the hot gas into stars and that is most efficient in low massive systems. In the local universe, the AGN balance the cooling thus the slope is closer to the gravitational-only model.

[**$M-{T_{\rm mw}}$ and $M-{T_{\rm sl}}$ relations.**]{} The slopes of both temperature relations, $M-{T_{\rm mw}}$ and $M-{T_{\rm sl}}$, drop at high redshifts. The values $\beta_{{T_{\rm sl}}}$ at $z=2$ are reduced with respect to the $z=0$ slope by $\sim23\%$, $\sim25\%$, and $\sim14\%$ in the [[[nr]{}]{}]{}, [[[csf]{}]{}]{}, and [[[agn]{}]{}]{} runs, respectively.

![image](fig5a){width="31.00000%"} ![image](fig5b){width="30.00000%"} ![image](fig5c){width="30.00000%"} ![image](fig5d){width="31.00000%"} ![image](fig5e){width="30.00000%"} ![image](fig5f){width="30.00000%"}

For either temperatures, the decrease in the $M-T$ slope does not strongly depend on the ICM physics. This indicates that the origin of the variation is due to macroscopic events, linked to the global evolution of the clusters.

To facilitate the explanation, we plot in Fig. \[fig:4\] the $M-{T_{\rm mw}}$ relation for the [[[nr]{}]{}]{} clusters at $z=0$ and $z=2$. As expected, high-redshift clusters are not thermalized yet and, therefore, they typically exhibit a lower value of temperature at fixed total mass, i.e. for $E(z) M < 2 \times 10^{14} h^{-1} M_{\odot}$ the $z=2$ temperatures (red points) are systematically on the left side of the $z=0$ ones 