Robert Baratheon uses a war hammer. Was that a common weapon for a high born knight?

Not uncommon, especially later on when plate armor got really really good. If you can’t cut through armor with your sword, then hitting someone really really hard at a single point and trying to buckle the armor or at the very least break some bones is a good idea. 

But I do want to point out, warhammers are a lot smaller IRL than in fantasy genre works. Let’s take Robert Baratheon’s replica for example:

While this is very visually impressive, it’s highly impractical. That giant face distributes the force of a blow across a pretty wide area, which is the opposite of what you want. It’s also a two-handed weapon, which makes it even heavier than it needs to be, and makes it quite difficult to use from horseback.

Here’s what a real warhammer from medieval Europe looks like:

Not that different from something you might find in your toolbox, or at the very least in a climbing supply store. It’s got a small head, because you want to concentrate the force of the blow into as small an area as you can, and it’s also got a pick head as well, because once you’ve knocked your opponent down, it’s easier to try to puncture their armor and really cause some damage. And it’s one-handed, because you need the other hand free for reins, a shield, etc. And it’s not that heavy, so you can keep swinging it all day long.