high redshift implies that it is extremely rare to grow a massive object via slow accretion. To summarize, we expect that while small clusters are not thermalized yet, the big ones can be already located in the $z=0$ scaling relation causing a shallower relation slope.

The change in slope is less prominent in the [[[agn]{}]{}]{} runs. In this circumstance, the gas of the smallest systems at $z=2$ is warmer because of the recent and intense AGN feedback activity. The phenomenon brings the smaller objects closer to the $z=0$ relation and reduces the amplitude of the slope evolution.

[**$M-{Y_{\rm X}}$ relation.**]{} The slope of the $M-{Y_{\rm X}}$ relation is constant until $z=1$ and then shows a mild decrease ($\leq 10\%$) for all the ICM runs consistent with previous results by [@sembolini.etal.2014] and by [@pike.etal.2014]. The origin of this variation can be understood by decomposing the $\beta_{{Y_{\rm X}}}$ slope into the two slopes of the $M-{M_{\rm g}}$ and $M-{T_{\rm sl}}$ [@2014MNRAS.437.1171M]: $$\beta_{{Y_{\rm X}}}=\frac{1}{1/\beta_{{M_{\rm g}}}+1/\beta_{{T_{\rm sl}}}}.$$ The complete derivation is presented in Appendix A. The mildness of the $\beta_{{Y_{\rm X}}}$ deviation is generated by the fact that none of the ICM physics runs present a coincident strong variation in both $\beta_{{M_{\rm g}}}$ and 