Looks like the centre piece of wood is for structural support. Leave it where it is. Find a plan B.

That center stile is part of the cabinet structure and was built as part of the cabinet construction. As a landlord I'd be furious if a renter removed or especially cut out part of the cabinetry, and I would definitely be keeping the security deposit. Have you considered finding a narrow cat litter box that could fit into the cabinet?

It'd probably be a pain, but can you get the litter box in sideways and then set it up in the cabinet and clean it in place?

Place masking tape over the right side of the bottom rail (if you're right handed). Place a Japanese pull saw on the tape and sever the bottom stile joint. After it's cut wriggle the top joint free. Throw it into the back of the cabinet for safekeeping. Hot melt glue it back in place when you're ready. The severed joint is not at eye level and will go back together inconspicuously.

Does the cat really go into a box inside a vanity cabinet? I feel like the cat deserves a bit better than that.

are you going to remove a door or keep one open for cat to get in and out?

You can ask your landlord to make the change for you. Pay them for a handyman to make the change and a deposit for putting it back together. You may have to replace the vanity and pay for installation as well if the top sags during your time there. If the bottom of the vanity is soiled by cat urine, you may end up paying for a new vanity and installation anyway. The upshot is that it would be better to find another place for the litter box.

IMO that stile is part of the cabinet strength so no . BTW cutting it out will not be the answer since when you add it back it will require filling and refinishing to make it look right again IMO a main bathroom is the worst place for litter box I like the laundry room best since I don’t have to sit and smell cat pee which BTW is awful if you are not a cat lover.

I suspect that stile is also support for the top, and removing it, the entire thing could cave in.

end table/night stand!

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Ikea lack table used to disguise litter box: https://www.youtube.com/watch?v=FEB6nFDA4sk

It's a cabinet, not a 17-story building. They stile is a door divider, nothing more.

get a smaller litter box

I have a frameless cabinet pantry with the center stile screwed to one of the doors. It's annoying because you need to close that door first but it will cover the gap.

Start lining your litter box with Johnny Cats or other litter liner. Then you won't have to pull the box out every time, just the liner and then put another in. It will be a bit of a PITA, but you love your kitty, right?

Respectfully disagree with the suggestion that you can remove it and glue it back in place when you move out. It would be ok to do this in a house you own, but not a house someone else owns. Just because you can doesn’t mean you should.