Unless the pricing is different for Hyperscale Serverless (which I have never used myself), the activity has nothing to do with it. What matter is whether the database is loaded on to a server or not.
To get benefit from Serverless, you should stop the database when you don't use it, or have it set to autopause. Here you have to be a little careful so that you don't have something that pings the database regularly.
Note that once pause, it takes some time for the database to come back, since "pause" means that the database is rolled out to a disk somewhere without any connection to SQL Server.