I agree with Dawnben about having the cat checked out by a vet...as this is unusual behavior.
Also, I had a cat named Ashley, and for years I used the same kitty litter in her box. But one year she started to avoid it and go elsewhere in the house...thankfully, it was the tub...so it was easy to clean up and disinfect.
Ashley had a history of feline urinary problems, crystals that build up in the bladder. My vet told me they were probably caused by the ash and magnesium fillers that are added to cat foods. But I had, had her on special foods to help prevent the infections...so I didn't think that was the problem. I still had her check out though and her urine tests came back as being normal.
First I thought, maybe she didn't like where her litter box was at for some reason....like maybe she got startled by the dogs while she was using it and that turned her off from it... So I moved the box to a different area of the house....that didn't work. So I put it back where it was to begin with.
I had been using kind of a course textured, scented, non clumping kitty litter...and as I said above, had used it for years. I also cleaned her kitty litter box once a day and scrubbed the thing down about every month and a half and replaced the whole box about once a year, since plastics tend to soak up odor and after a while, even cleaning it with a bleach solution doesn't get to the source of the odor as it's into the plastic too deep.
I thought about it some more, and thought that maybe because she was getting older (she was 14 at the time) that something in the kitty litter was bothering her. So I started to experiment with other kitty litters.
I first tried the odorless version of that kitty litter I had used in her box for years, and that wasn't it, she would use the box sometimes but was still going in the tub more often than not.
So, I tried a scented, clumping litter that had a really fine soft 'sandy' feel to it, and Ashley went right back to using her litter box. She never went into the tub again after that.
Apparently, as she got older the texture of the kitty litter bothered her feet, and perhaps she became more sensitive to the scent in her old scented litter and also because it didn't clump, maybe she started to dislike the moist feel at the bottom of the kitty box when she got done. I don't know...but was sure glad when I finally found something that got her out of my tub and back into the kitty litter box.
I hope you get your cat's problems figured out.
Stormy