  A class of Aubry-Andr\'e-Harper models of spin-orbit coupled electrons
exhibits a topological phase diagram where two regions belonging to the same
phase are split up by a multicritical point. The critical lines which meet at
this point each defines a topological quantum phase transition with a
second-order nonanalyticity of the ground-state energy, accompanied by a linear
closing of the spectral gap with respect to the control parameter; except at
the multicritical point which supports fourth-order transitions with parabolic
gap-closing. Here both types of criticality are characterized through a scaling
analysis of the curvature function defined from the topological invariant of
the model. We extract the critical exponents of the diverging curvature
function at the non-high symmetry points in the Brillouin zone where the gap
closes, and also apply a renormalization group approach to the flattening
curvature function at high symmetry points. We also derive a basis-independent
correlation function between Wannier states to characterize the transition.
Intriguingly, we find that the critical exponents and scaling law defined with
respect to the spectral gap remain the same regardless of the order of the
transition.
