Do the Ernie cobalt flat wound strings tend to lose their action?

What exactly do you mean by action? Are you talking about the strings losing tension, or what action really means - the height from the frets?

If it's the latter, then string age doesn't matter at all. Usually what happens is either that the string gauge is different - causing the neck relief to change, something which will affect the action (string height); or the bridge or bridge saddles have moved when you changes strings (some times the action changes over time on some instruments, simply because the bridge or bridge saddles move).

If you have changes string gauge/string tension and it affects the neck relief too much, you either need to go back to the original gauge (which have the same overall tension), or get a tech to adjust the truss rod slightly for you (please do not attempt this yourself). Some times it is enough to raise the action at the bridge a fraction.

Edit:

I see, you're talking about tone. Action is string height and is something completely else than what you're asking about here.

Are you referring to brightness when you say "slap tone"? From your initial question and your follow up, it is very hard to say if the strings are becoming too dead/dull or they have too much brightness. You really isn't making this easy for us here :-)

If there is too much brightness/sparkle, the easiest way is to roll off some treble on the bass or the amp. If you feel these strings become dull, then the obvious thing is to go for regular round wound strings.

But again - maybe you're talking about something completely else...

Yeah I've read your comment - "action" is the height of the stribgs above the fretboard, it has nothing to do with tone.

All strings go dead over time. New strings are very bright. To me, this brightness dissappears very quickly and there is a period when the strings just sound more "normal", even and balanced. Then they go dead.