You can buy protective passport pouches, these hang on your neck and have compartments for other travel "stuff", like museum or railpasses. The price is under $10.
If you leave your pp in your hotel room, it could be stolen; also, you never know when you will have to show the pp.
In a thread elsewhere on FT, some people made negative comments (stylewise) about people using a pp pouch...I think they are super convenient.
+1
I have such a pouch . I use it on international travels and if it is not a brand na,e chain hotel and no safe, I carry it. Just wear on your neck under the shirt.
This is also good at TSA line up juggling too many items (DL/PP, ticket - BP, shoes, laptop etc.