  We investigate the order of the topological quantum phase transition in a two
dimensional quadrupolar topological insulator within a thermodynamic approach.
Using numerical methods, we separate the bulk, edge and corner contributions to
the grand potential and detect different phase transitions in the topological
phase diagram. The transitions from the quadrupolar to the trivial or to the
dipolar phases are well captured by the thermodynamic potential. On the other
hand, we have to resort to a grand potential based on the Wannier bands to
describe the transition from the trivial to the dipolar phase. The critical
exponents and the order of the phase transitions are determined and discussed
in the light of the Josephson hyperscaling relation.
