Synchronized aggregate signature (AS) can aggregate multiple individual signatures into one signature, when the individual signatures are generated at the same period. Although several synchronized AS schemes have been proposed, all of them are not quantum computer resistant. In this paper, we discuss a lattice-based synchronized AS that is secure in the standard model and the certified-key model. The proposed scheme is based on the multi-signature scheme proposed by Fleischhacker, Herold, Simkin and Zhang. We convert their multi-signature scheme into the synchronized AS scheme by separating a public key into two parts and applying the homomorphic vector commitment to each part. Our technique can handle hash values of messages as coefficients of homomorphic operation even if a message is different for each user, and then we can aggregate individual signatures for different messages into one signature. Our result is not only the first lattice-based synchronized AS but also the first lattice-based AS secure in the standard model.