time-dependent couplings

Imagine that the \(\Lambda\)-term in Einstein’s field equation was time dependent, \(\Lambda(t)g_{\mu\nu}\). why does this imply that the gravitational constant needs to be time-dependent as well?

One could take the derivative of the entire field equation, keeping in mind that both \(G_{\mu\nu}\), \(T_{\mu\nu}\) and \(g_{\mu\nu}\) are zero, such that \(8\pi \nabla_\mu G T^{\mu\nu} + \nabla_\mu\Lambda g^{\mu\nu} = 0\). The two constants, \(G\) and \(\Lambda\) are scalars, such that \(8\pi\partial_{t}G T_{t\nu}+ \partial_{t}\Lambda g_{t\nu}\) in a preferred frame, implying the time dependence of \(G\).