The cross sections was measured for coherent double neutral-pion photoproduction on the deuteron using the tagged photon beams from the 0.9 and 1.2 GeV primary electron beams at the Research Center for Electron Photon Science, Tohoku University, Japan. The four-momenta of the four photons from the two $\pi^0$ decays and deuteron are measured with the FOREST detector, and the $\gamma d\to \pi^0\pi^0d$ reaction was identified using 6-constraint kinematic fit (energy and three-momentum conservation, and two $\pi^0$ masses). The total cross section is found to be $18.4\pm 9.2$~nb for the $d^*(2380)$ dibaryon resonance at $W_{\gamma d}=2.37$~GeV, which is a candidate of the isoscalar $\Delta \Delta $ quasi-bound state ${\cal D}_{03}$, in the $\gamma d \to \pi^0\pi^0 d$ reaction. The measured excitation function is consistent with the existing theoretical calculation for this reaction. At high incident energies, $\pi^0d$ invariant-mass distributions show a peak at approximately 2.15 GeV, which can be attributed to a $N\Delta$ quasi-bound state ${\cal D}_{12}$. In this contribution, the properties (mass, width, and decay angular distribution) of this peak is presented.