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

The Three Hundred Project: dissecting the fundamental plane of galaxy clusters up to $z=1$