Summary: We present a study of the evolution of entanglement entropy of matter and geometry in quantum cosmology. For a variety of Gaussian initial states and their linear combinations, and with evolution defined with respect to a relational time, we show numerically that (i) entanglement entropy increases rapidly at very early times, and subsequently saturates to a constant non-zero value, and (ii) that the saturation value of this entropy is a linear function of the energy associated to the quantum state: \(S_{\mathrm{ent}}^{\psi}=\gamma\langle \hat{H}\rangle_{\psi}\). These results suggest a remnant of quantum entanglement in the macroscopic Universe from the era of the Big Bang, independent of the initial state parameters, and a ‘First Law’ associated with matter-gravity entanglement entropy in quantum gravity.