[e.g. @nord.etal.2008]. The history of the $L-T$ relation has been equally important because from its first determination it was clear that this relation provides information on the physics of the cluster core [e.g., @fabian.etal.1994] and the phenomena of feedback by stars or AGN [e.g., @maxim98; @2012MNRAS.421.1583M].

Over the last decade, a significant number of realistic hydrodynamical simulations enabled the study of scaling relations (@2011ASL.....4..204B for a review). A special attention was dedicated to the effects of feedback from stars [@nagai.etal.2007] and AGNs [@short_thomas_2009; @puchwein.etal.2008; @2010MNRAS.408.2213S; @2014MNRAS.438..195P; @lebrun.etal.2014; @pike.etal.2014], to the evolution of the relations up to $z\sim1$ [@2012ApJ...758...74B; @2011MNRAS.416..801F; @planelles.etal.2016; @lebrun.etal.2016], and to develop a theoretical framework to exploit the simultaneous analysis of multiple signals [@2010ApJ...715.1508S; @evrard.etal.2014]. This paper, based on realistic models of ICM quantities obtained through a new BH accretion model, AGN feedback and hydro scheme, extends the analysis to $z=2$ in vision of up-coming observational results. Most importantly, this work analyzes a simulated sample that recently proved to naturally form in a cosmological context cool-core (CC) and non-cool-core (NCC) clusters [@2015ApJ...813L..17R]. The diversity of the core entropy level in the centermost regions is mostly ascribed to the dynamical history of the system and to its overall AGN 