There are some pieces of music that work very well when played without keeping a consistent tempo, in popular music I'm thinking of songs such as Colorblind by Counting Crows or the intro to Shanghai by Andy McKee. These are songs that speed up and slow in a sort of rhythm, almost like drawing breath.
Is there a name for this in music? The closest I could find was
A piacere, but as far as I could tell that really means that the musician picks a tempo when they start to play, not that the tempo should regularly change throughout the piece. Perhaps I'm wrong here