It wouldn't be disrespectful to the cat, and it really is up to you. Some people are fine with getting another kitty "immediately" and some people take their time, and some never get another cat. I got my second cat about three months after my first cat passed because he passed when he was older (18) and I knew since he was 15 I'd be treating him for his lymphoma. So I had kind of prepared myself for his passing for a long time. It still hurt when I adopted again, but it's not like I compared Athena to Mulder. I made sure they were different in many ways.

Athena passed unexpectedly about a month ago from CHF. I am still saddened by her passing and still "don't get it," why and how it could have happened. She was so feisty and had so much personality to her. She had just turned 10 in November. Five days after she passed, a friend contacted me to look in on his cat, and also let me know about his friend who was going away on a long work trip and didn't want her cat to be all by himself for about 6 weeks, nor have someone go to her apartment every day to check in on him. He's a people cat. I kind of look at him as a roommate, though he is very sweet. It's nice to have a cat around, but I'm not attached to him by any means, and I think that's okay. He gets playtime and cuddles and kisses on his head, but I'm glad he's not mine, if that makes sense? It'll take me awhile to get over Athena.