I'm trying to find a Japanese book titled something like "When all the cats in the world had died" (translation may be inaccurate). I tried Googling that English title-- no luck.
The story is about a man awaiting his death. The devil offers him one extra day to live, for each important thing he gives up doing everyday.
The book was most probably published in 2013, is a short story and the cover image is grey colored with a cat picture on it.