Using 2.92 fb−1 of electron-positron annihilation data collected at s√=3.773 GeV with the BESIII detector, we report the results of a search for the flavor-changing neutral current process D0→γγ using a double-tag technique. We find no signal and set an upper limit at 90% confidence level for the branching fraction of B(D0→γγ)<3.8×10−6. We also investigate D0-meson decay into two neutral pions, obtaining a branching fraction of B(D0→π0π0)=(8.24±0.21(stat)±0.30(syst))×10−4, the most precise measurement to date and consistent with the current world average.