  We explore higher order topological superconductivity in an artificial Dirac
material with intrinsic spin-orbit coupling, which is a doped $\mathbb{Z}_2$
topological insulator in the normal state. A mechanism for superconductivity
due to repulsive interactions -- $\mathit{pseudospin}$ $\mathit{pairing}$ --
has recently been shown to naturally result in higher-order topology in Dirac
systems past a minimum chemical potential \cite{Li2021}. Here we apply this
theory through microscopic modeling of a superlattice potential imposed on an
inversion symmetric hole-doped semiconductor heterostructure, known as
hole-based semiconductor artificial graphene, and extend previous work to
include the effects of spin-orbit coupling. We find that spin-orbit coupling
enhances interaction effects, providing an experimental handle to increase the
efficiency of the superconducting mechanism. We show that the phase diagram of
these systems, as a function of chemical potential and interaction strength,
contains three superconducting states -- a first-order topological $p+ip$
state, a second-order topological spatially modulated $p+i\tau p$ state, and a
second-order topological extended $s$-wave state, $s_\tau$. We calculate the
symmetry-based indicators for the $p+i\tau p$ and $s_\tau$ states, which prove
these states possess second-order topology. Exact diagonalisation results are
presented which illustrate the interplay between the boundary physics and spin
orbit interaction. We argue that this class of systems offer an experimental
platform to engineer and explore first and higher-order topological
superconducting states.
