this book, as all Heinleins' works, is an absolute gem and a joy to read; the sort of book you cannot put down, and when you get to the end you're sad because there are no more pages.
So good we named one of our cats Pete...read the book and find out why this is relevant!
The door into summer is a cat's version of "the grass is always greener...", and it refers to the fact that his cat- as all cats - will go to the front door, reject this means of exit because it is raining and go to the back door, hoping to find it sunny there, in other words, wanting to find the door into summer. The whole book is full of typical Heinlein hypotheses...what ifs.
If you don't buy this and read it you really are missing out on one of the best novels ever written, whether you like sci-fi or not.
(NB Our other cat is called Spangle; Michael Marshall Smith readers will understand this one...!)