The Three Hundred project: dissecting the Fundamental Plane of galaxy clusters up to <i>z</i> = 1

We perform a systematic study of the recently discovered fundamental plane galaxy clusters (CFP) using ~250 simulated from The Three Hundred project, focusing on stability against different temperature definitions and its dependence dynamical relaxation state clusters. CFP is characterised by $T\propto M_\mathrm{s}^\alpha\,r_\mathrm{s}^\beta$, defined with gas ($T$) characteristic halo scale radius mass ($r_\mathrm{s}$ $M_\mathrm{s}$) assuming Navarro-Frenk-White description. explore two weighted temperatures, namely mass-weighted spectroscopic-like in three radial ranges. project at $z=0$ lie thin whose parameters ($\alpha,\beta$) dispersion ($0.015-0.030$ dex) depend definition. for temperatures closer to virial equilibrium expectation ($\alpha=1,\beta=-1$) smaller dispersion. For measured within $500h^{-1}$ kpc, resulting deviates most shifts towards similarity solution secondary infall model ($\alpha=1.5,\beta=-2$). Independently definition, we find that $z=1$ relaxed form similar expectation, unlike disturbed exhibiting stronger evolution. Only systems formed over last 4 Gyr present solution. All these findings are compatible obtained CLASH subsample excluding hottest $T_\mathrm{X}>12$ keV.