Perhaps one of the most important results of the whole of Classical Mechanics is that the volume occupied by an ensemble in the phase space remains constant in time. Another very interesting result is that it is also invariant under canonical transformations (as the Jacobian of the canonical transformation is generically unity).
Both these results compel me to think that there should be some extremely simple and deep physical meaning to the volume of the phase space. Intuitively, it seems to me that since the number of points occupied in the phase space is just the number of microstates of the ensemble, the volume of the phase space also denotes just the number of microstates of the ensemble up to some (not-so-clear-to-me) scaling factor. My question, thus, is that what is this scaling factor? Is it just arbitrary (i.e., to be decided by some choice of units etc.) or is there a definite scaling factor?