For rubbers, what we call hardness is a measure of how deep an indenter is pressed into the rubber, given a certain force or energy... but when the indenter is removed, there is typically no permanent mark, as there would be for a metal. Hence, the hardness is a measure of elastic deformation vs load, much as the various forms of stiffness (Young's, shear, bulk modulii) are. In that sense, I believe that they are directly related. However, you also have to consider the degree of crosslinking, fillers, additives, chain branching, degree of polymerization, whether there is co-polymerization, and so on.... So I think that we can safely say there is a direct correlation, but that it is not well defined because of all the additional factors. For isotropic materials, we know that Young's Modulus and Shear modulus are related, using a Poisson's ratio, but are all rubbers isotropic? ..... in reality due to chain allignment, they are probably not, although we assume they are.
I would think that if you were to thoroughly search the work on this area, that some fairly good correlations between shore hardness and bulk modulsus, for example, within sub-groups of rubbers, that have well defined levels of crosslinking, branching, DP, low crystallinity, and so on, could be found.
try scirus.com for a search engine and good luck!