I'm using Linux (Ubuntu 18.04)

Are you possibly using Sync?

Firefox uses the Password Manager to store the Sync credentials, so you need to enter the master password when Firefox starts to unlock the Password Manager.

Yes, I'm using sync and I understand the way it works. But it is a bit annoying to start Firefox on search engine page, begin to type a request and then the main password window pops up! It would be better for users to ask for main password immediately on startup, like Thunderbird does.

Do you perhaps have several tabs set for auto open ? OR several (potentially long running modified prefs/settings) in your profile that would delay this prompt ? the workflow logic (when using either sync or master password ) is to use the same flow as thunderbird, if you are noticing that long of a delay in firefox only, I'd recommend consulting your prefs.js usre.js or other extensions/add-ons or even stale profile caches (including those in your sync profile.)