The game essentially packages a character's maximum velocity, acceleration potential, and agility into a commodity called "speed". This makes sense because these attributes are impossible to meaningfully compare across a variety of creatures of different sizes, strengths, body configurations, etc.
The Mobile feat (PHB, p. 168) essentially adds 10 "speed points" to a character's movement budget to reflect an increase in all movement abilities.
Thus a fighter with a pre-feat budget of 30 "speed points" must spend 15 (50%) of those to stand from prone. The question is: does it still cost him 15 "speed points" to stand post-feat? Or must he now spend 20 "speed points" (5 more) to do the exact same thing (more slowly) when, post-feat, he is presumed faster and more agile?
Best Answer
Yes, it's halved
And
So mobile increases your speed, and prone costs you half your speed. If you have 30 speed and add 10 from mobile you get 40, if you stand up from prone that turn you have 20 left since standing up costed you 20. Nothing contradicts this general ruling, so it's RAW and (most likely) RAI.
If you have a hard time wrapping your head around this think it this way: mobile allows you to run faster, but not to move faster. So when you are standing up you still consume half your movement to stand up, since you can't stand up faster than someone without mobile, but once you stand up and start running you still can run faster than someone without the feat since your running speed is higher, just not as much distance as normally since you only have half the time to run.
However, as @PixelMaster said, you could get the Athlete feat to reduce the cost of standing up from prone to only 5 feet of movement, thus benefitting from the full 10 speed of the Mobile feat.