We will introduce and discuss a `moduli space' of metric structures on hyperbolic groups that can be seen as an extension of Teichmüller Space or Outer Space. This space contains metrics of various flavours coming from random walks, actions on hyperbolic metrics spaces, Anosov representations and more. After equipping our space with a natural metric, we will look at its geometric properties and see how, by studying these properties, we can learn about and compare metrics coming from different geometric settings. This is based on joint work with Eduardo Reyes.