Difficult terrain does not affect standing up from prone.
The rules for difficult terrain state:
You move at half speed in difficult terrain— moving 1 foot in Difficult Terrain costs 2 feet of speed
"Move" and "moving" here are referring to the rules for movement and position:
On your turn, you can move a distance up to your speed. You can use as much or as little of your speed as you like on your turn, following the rules here.
Your movement can include jumping, climbing, and swimming. These different modes of movement can be combined with walking, or they can constitute your entire move. However you're moving, you deduct the distance of each part of your move from your speed until it is used up or until you are done moving.
The distance travelled when standing up from prone is 0 feet, so you "move" 0 feet when standing up from prone. Since you move 0 feet when standing up from prone, difficult terrain does not impose any additional penalties when standing up.
It is worth noting that this question was answered in version 1.14 of the Sage Advice Compendium, but was removed without comment for version 2.0:
Is standing up from prone considered moving? Standing up costs movement but moves you nowhere. When the game refers to you moving, it means moving some distance. It doesn’t mean making a gesture or standing up in place. To move while prone, you crawl or use magic (PH, 191).