Since hardness measurement involves only surface of the material, bulk density of the material does not have any explicit relation with the hardness. Materials are usually subjected to microstructural and/or chemical changes in the surface by carbiding, nitriding and other processes, many times by cold working.
The structure of the surface of the material is almost always different from the structure inside the bulk, it would be a very complicated and complex relationship, if anyone ever tries to establish it and may not prove that beneficial also
