Formation redshift from halo mean density ($1+z_f=[3M/(4\pi R^3{\Delta }_f{\rho }_0){]}^{1/3}$)

In a spherical-overdensity halo convention, $3M/(4\pi R^3)={\Delta }_f{\rho }_0(1+z_f{)}^3$, so $1+z_f$ follows from the displayed cube root. At fixed measured $M,R,{\rho }_0$, changing the adopted overdensity changes $1+z_f\propto {\Delta }_f^{-1/3}$. It does not change the hydrostatic temperature estimate from the same measured mass and radius. A redshift inferred this way is a model formation epoch, not a direct age measurement.