 Solution

At fixed $\eta$, the recombination temperature is determined by atomic physics and the Saha equation, not by today's CMB temperature. Under the stated approximation,Since $1+z_{\mathrm{dec}}=T_{\mathrm{dec}}/T_0$,and the same ratio holds for the large redshifts themselves to excellent accuracy. The comoving distance ${\chi }_{\mathrm{dec}}={\int }_0^{z_{\mathrm{dec}}}dz/H(z)$ can nevertheless be held fixed by adjusting the late-time expansion history, for example $H_0$, ${\Omega }_{m,0}$, spatial curvature, or dark-energy density and equation-of-state parameters.