activity. We expect that in our simulated clusters the balance between radiative cooling, which forms stars, and AGN feedback, which heats the gas, is realistically reached as the systems evolve in and interact with the embedded cosmological environment. Therefore, it is compelling to study the relation between the ICM properties in an epoch right after the maximum level of the AGN activity.

The paper is arranged in the following way: in Section 2 we provide a short description of the simulated sample and refer to companion papers for a more detailed explanation of the code. Section 3 presents the computation of ICM structural quantities, the mass-proxy relations, the luminosity-based relations, and fitting methods. In Section 4, we examine the validity of our simulated data by comparing to observations at low ($z\leq0.25$) and intermediate ($0.25\leq z \leq 0.6$) redshifts. Section 5 is dedicated to explore the theoretical evolution of scaling relations from $z=0$ to $z=2$. Finally, a summary of the obtained results and conclusions is given in Section 6. All the quantities for the scaling relations are evaluated at $R_{500}$ defined as the radius of the sphere whose density is 500 times the critical density of the Universe at the considered 