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Hello. I'm sorry to hear about your cat.
It sounds like he may have Vestibular Disease. This is a condition where the balance is off. It's similar to vertigo in people. It typically comes on suddenly, then goes away slowly over 7-10 days.
I'm glad you checked the temperature of the rear legs. A saddle thrombus or clot to the back legs is also possible. Those cats are usually paralyzed in the rear legs, very painful and the feet/ paw pads are cold and a blue-purple color.
Either way, I would get him to a vet. You could either go to an ER vet tonight or your regular vet in the morning. Getting this diagnosed and treated is important. If Vestibular, we typically use anti-motion sickness meds (Meclizine) +/- antibiotics if we think it's an inner ear infection as the cause.
I hope this helps, let me know if you have any other questions.
If it's Vestibular, we usually see rapid back and forth movement of the eyes that look like twitching. If it's the whole body, it could be a discomfort things.
I would go now if you can, just to error on the side of caution.