The Last Policeman, by Ben H. Winters
The speculative premise of this book is what drew me to it: an asteroid on a crash course with Earth and how that affects law and order. Those elements also turned out to be my favorite as I read--not just how the government and law enforcement would change with a world-changing disaster a year away, but how people live more generally. The collapse (essentially) of the economy. Increased suicides. Increased drug and alcohol use.
Henry Palace, the first person narrator, is a detective, one of those guys meant to be one. He's meticulous and one of the only members of the police force in Concord, New Hampshire, to follow a suspected murder case (as opposed to the suicide it's assumed to be) as if it still matters. The book can read a bit like a noir detective story, which may or may not be your cup of tea (it's less my cup). But I liked him and his diligence.
The case he works takes lots of turns, some of which I saw coming, many of which I didn't. Sometimes the story is structured so that Palace knows what's going on, but there's a bit of a tease before the reader finds out; that was a little annoying to me, though you're not strung along for too long.
I liked the book; it was enjoyable enough. I saw that there's at least one more book set in this world, though I didn't love this novel enough to want to check out the next in the series. Sometimes one book's enough.