Why is my 10-year-old cat having diarrhea and litter box accidents?

I am sorry to hear about your kitty! If the food appears different than the one he normally eats it is possible that perhaps there is a different additive or food coloring that has upset his stomach (or is it possible it is a different flavor even though the brand is the same). We can see some pets have GI upset from quick changes in diet. That being said, we can also see diarrhea from other causes, including GI parasites, bacterial overgrowth, inflammatory bowel disease, as a symptom of systemic illness, and more. If he is still energetic with no vomiting you could try getting a probiotic from your vet or the pet store for him which can sometimes help with diarrhea and feeding something bland for a few days (such as plain white rice and shredded boiled chicken or lean ground chicken) or picking up the food you normally get at the pet store and trying this for a few days. If he is having urgency so often he isn't making it to the litterbox, however, it is quite possible he has colitis http://www.veterinarypartner.com/Content.plx?P=A&S=0&C=0&A=677 and will likely need prescription medication to get the diarrhea under control, so the fastest solution to eliminating the problem is to have your vet examine him!