Here[1], the density is said to be as low as 0.03 gm/cc which is the same as kg/liter. That gives a multiplicative factor of 33 over the above calculations, i.e. this could be 33% of the energy density of lithium ion batteries. Not bad.
Energy per liter also is an important factor. Your mobile would float, but it also would be the size of your lower arm. Maybe, you should wear your battery as a jacket or a pair of trousers?