We carry out a systematic study of the recently discovered fundamental plane of galaxy clusters (CFP) using a sample of ~250 simulated clusters from the 300th project, focusing on the stability of the plane against different temperature definitions and its dependence on the dynamical relaxation state of clusters. The CFP is characterised in the form of $T propto M_s^alpha r_s^beta$, defined with the gas temperature ($T$) and the characteristic halo scale radius and mass ($r_s$ and $M_s$) assuming an NFW halo description. We explore two definitions of weighted temperatures, namely mass-weighted and spectroscopic-like temperatures, in three radial ranges: [0.1, 1.0]$r_{200}$, [0.15,1.0]$r_{500}$, and [50,500]$h^{-1}$ kpc. We find that 300th clusters at $z=0$ lie on a thin plane whose parameters ($alpha, beta$) and dispersion (0.015--0.030 dex) depend on the gas temperature definition. The CFP for mass-weighted temperatures is closer to the virial equilibrium expectation ($alpha=1, beta=-1$) with a smaller dispersion. When gas temperatures are measured inside 500$h^{-1}$ kpc, which is close to the median value of $r_s$, the resulting CFP deviates the most from the virial expectation and shifts towards the similarity solution for a secondary infall model ($alpha=1.5, beta=-2$). Independently of the temperature definition, we find that clusters at $z=1$ form a CFP similar to the virial expectation. At all epochs, the CFP remains well defined throughout the evolution of the cluster population. The CFP of relaxed clusters is always close to the virial expectation, with a milder evolution than for the unrelaxed case. We find that only systems formed over the last 4 Gyr have a CFP that is closer to the self-similar solution. All these findings are compatible with the CFP obtained for a CLASH subsample excluding the hottest clusters with $T_X>12$ keV.
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