  We investigate the quantum criticality and universality in
Aubry-Andr\'{e}-Harper (AAH) model with $p$-wave superconducting pairing
$\Delta$ in terms of the generalized fidelity susceptibility (GFS). We show
that the higher-order GFS is more efficient in spotlighting the critical points
than lower-order ones, and thus the enhanced sensitivity is propitious for
extracting the associated universal information from the finite-size scaling in
quasiperiodic systems. The GFS obeys power-law scaling for localization
transitions and thus scaling properties of the GFS provide compelling values of
critical exponents. Specifically, we demonstrate that the fixed modulation
phase $\phi=\pi$ alleviates the odd-even effect of scaling functions across the
Aubry-Andr\'{e} transition with $\Delta=0$, while the scaling functions for odd
and even numbers of system sizes with a finite $\Delta$ cannot coincide
irrespective of the value of $\phi$. A thorough numerical analysis with odd
number of system sizes reveals the correlation-length exponent $\nu \simeq
1.000$ and the dynamical exponent $z$ $\simeq$ 1.388 for transitions from the
critical phase to the localized phase,suggesting the unusual universality class
of localization transitions in the AAH model with a finite $p$-wave
superconducting pairing lies in a different universality class from the
Aubry-Andr\'{e} transition. The results may be testified in near term
state-of-the-art experimental settings.
