It looks like you can set up each route in “Components/HTTP Routes”, but I don’t see an easy was to set a catch-all. Perhaps overriding the static_sites/error_document key would do it (as referenced here:
https://www.digitalocean.com/docs/app-platform/references/app-specification-reference/ )
but I’m assuming that is responding with a 404, which wouldn’t be good.

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Mod edit: we have since added support for “Catchall Documents”. In your Static Site settings, you will have the option to choose a file (e.g. index.html) that all inexistent paths are routed to with a 200 OK status code.