Pay rent

Usually "pay rent on". "pay a rent on" is not actually incorrect though.

A: Do your rent or do you own?
B: I rent.
A: What?
B: I rent an apartment.
A How much rent do you pay for your apartment?
B: $400 a month.
A: isn't that a lot?
B: It's cheaper than paying rent on a house.

Thank you for your answers!

I'm confused why it's "for" in this example

deadratA How much rent do you pay for your apartment?

but "on" in this one

deadratB: It's cheaper than paying rent on a house.

There are two contexts. The first is "paying" and the second is "renting." Generally, you use "for" when you exchange money for something. But for some reason, rent money goes "on" your abode. So it's simply a matter of what you're emphasizing in the conversation. If you switched the prepositions, both sentences would be fine.

As I understand this, you could say:

How much rent do you pay on your apartment?
It's cheaper than paying rent for a house.

Have I understood you correctly? Thanks!

Yes. "Pay for" is easily the more popular.