  Coarse geometry studies metric spaces on the large scale. The recently
introduced notion of coarse entropy is a tool to study dynamics from the coarse
point of view. We prove that all isometries of a given metric space have the
same coarse entropy and that this value is a coarse invariant. We call this
value the coarse entropy of the space and investigate its connections with
other properties of the space. We prove that it can only be either zero or
infinity, and although for many spaces this dichotomy coincides with the
subexponential--exponential growth dichotomy, there is no relation between
coarse entropy and volume growth more generally. We completely characterise
this dichotomy for spaces with bounded geometry and for quasi-geodesic spaces.
As an application, we provide an example where coarse entropy yields an
obstruction for a coarse embedding, where such an embedding is not precluded by
considerations of volume growth.
