>>I hear it even when I put the car in neutral and switch the engine off.<<
You said the tone doesn't vary with speed, but guessing the volume does?
A bad wheel bearing can drone like the engines of a B29 Flying Fortress (similar to this
https://www.youtube.com/watch?v=Tds7Onwz2-U), but it gets louder with speed and is not affected much by temperature.