I am running an app on ASA8 for years. Even a empty database (regarding data) has a size because of its data structure. This is also influenced by the page size of the database.
When you begin adding data the database might be able to fill in the data into the existing pages without the need to extend the file size for some time. But you should soon see an increasing file size of the database.
ASA writes new or updated data to a log file in the same directory as the database file (default option). The changes from this log file are fed into the main database when the database server executes a so called checkpoint. Check the server log messages - a message should inform you about checkpoints to happen. After a checkpoint the timestamp of the database file gets updated. Check this too.