  We perform a systematic study of the recently discovered fundamental plane of
galaxy clusters (CFP) using ~250 simulated clusters from The Three Hundred
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 by $T\propto M_\mathrm{s}^\alpha\,r_\mathrm{s}^\beta$,
defined with the gas temperature ($T$) and the characteristic halo scale radius
and mass ($r_\mathrm{s}$ and $M_\mathrm{s}$) assuming a Navarro-Frenk-White
halo description. We explore two definitions of weighted temperatures, namely
mass-weighted and spectroscopic-like temperatures, in three radial ranges. The
Three 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 gas
temperature definition. The CFP for mass-weighted temperatures is closer to the
virial equilibrium expectation ($\alpha=1,\beta=-1$) with a smaller dispersion.
For gas temperatures measured within $500h^{-1}$ kpc, 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 $z=1$ and relaxed clusters
form a CFP similar to the virial expectation, unlike disturbed clusters
exhibiting stronger evolution. Only systems formed over the last 4 Gyr present
a CFP that is closer to the similarity solution. All these findings are
compatible with the CFP obtained for a CLASH subsample excluding the hottest
clusters with $T_\mathrm{X}>12$ keV.
