There are lots of famous cats, from Morris to Grumpy Cat, and they all have one thing in common - they have no desire to be famous. They're in the public eye because their human wants them to be famous or make money off them, or both. This is wrong for a number of reasons, but the worst is that it compromises the well-being of the cat.
I live in Los Angeles, and I've worked on a large number of big budget film and television shoots. Everything from Transformers and Pirates of the Caribbean to Alias and My Wife & Kids. What they all have in common is extremely high-pressure environments and a serious lack of time. If you add an animal into the mix, your exacerbate those problems. Now the animal is expected to perform on cue for grumpy, impatient people who just want to finish the day and go home. This isn't a situation where an animal will be well-treated. In fact, despite the claims of the American Humane Association, most animals on set are treated quite badly. They're transported in cramped conditions, moved into spaces they don't know and can't understand, and then expected to perform on cue.
Cats are especially susceptible to becoming distressed in this sort of situation, unlike dogs who might even thrive there. Cats are highly territorial. When taken to an area that lacks familiarity and their scent markings, they can become agitated. Just how agitated really depends on the individual cat. Some cope with change better than others, such as Maine Coons, but almost all cats dislike new places the first time they're taken there. Regardless of personality, their reactions are either to hide from the world or to explore an area that is extremely unsafe for animals.
By the way, "American Humane Association" (AHA) and "The Humane Society of the United States" (HSUS) are different organizations. The AHA film & TV unit makes a significant amount of money from Hollywood productions that want to have their projects certified. Despite the AHA film tagline, "No animal was harmed", it seems that a number of animals have been harmed under the AHA's watch. In 2001, the Los Angeles Times ran an article on the AHA film & TV unit that stated, "the group has been slow to criticize cases of animal mistreatment, yet quick to defend the big-budget studios it is supposed to police." The article also cites specific incidents where animals were injured or killed on set with no repercussions from the AHA. (read the entire article here) The most recent accusations revolve around animal deaths on the set of The Hobbit.
Thankfully, CGI animals are becoming more prevalent in today's productions. I applaud the use of CGI as long as the animals portrayed aren't harmed. Sadly, dramatizing the violent death or mistreatment of a CGI animal can give some psychotic individuals the wrong idea.
You may be asking what you can do. Well, the first thing is to stop supporting productions that use live animals in this way. A film that uses cats is no different that a circus that uses elephants. Just don't buy a ticket. If you see an online video where a cat is frightened or tormented, note your disgust in the comments. We have to make the general public aware that animals are sentient beings and that we won't tolerate their mistreatment for human profit or pleasure.