When GETting some objects, you may have a versionTag in the response with a lastModifiedOn timestamp. However, this is not available at project level. It would be possible to reconstruct it by iterating over all objects within a project, but is more involved.
A simpler approach would be to use the internal commits database, as shown below:
This dataset has all the commits by date for all projects. It can be used in a Scenario to automate the identification of recently modified vs "stale" projects.
Hope it helps,