The documentation is very vague on how to customize the seekbar/progress indicator for the FLVPlayback component and Google searches return few hits on how to actually do it with the provided component. There’s a good blog post about it here, that actually explains the procedure. In case it ever disappears, here’s my own note on the subject.
Step 1: Start by creating a new layer between the assets and progress layers, and call it something, i.e. “play progress”.

Step 2: Create a new MovieClip that is as long as you want it to be when the video has completed, and name that MovieClip fullness_mc.

Step 3: Remove or hide (by making the layer a guide) the progress indicator MovieClip.
If you’re creating a custom FLV player, assign the seek bar to the playback component with this line of code:
player.seekBar = mySeekBar;
player being the playback instance and mySeekBar being the seek bar instance. And make sure not to add the code line twice, which made it not work for me.
And this is the final result:

Additional step: To make navigation quicker and to allow for the user to click on the seek bar to fast forward the video (instead of trying to hit the blue handle below the seek bar), create a new layer called hit area and add a MovieClip with the instance name hit_mc so that it covers the fullness_mc MovieClip. Set the alpha of the hit area to zero, and then the user can click to go forward and backward. In case you want a hand/button indicator, add this.buttonMode = true to the scripts layer.