  We investigate the stability of MnPb$_{2}$Bi$_{2}$Te$_{6}$ (MPBT), which is
predicted to be a magnetic topological insulator (TI), using density functional
theory calculations. Our analysis includes various measures such as enthalpies
of formation, Helmholtz free energies, defect formation energies, and dynamical
stability. Our thermodynamic analysis shows that the phonon contribution to the
energy gain from finite temperature is estimated to be less than 10~meV/atom,
which may not be sufficient to stabilize MPBT at high temperatures, even with
the most favorable reactions starting from binaries. While MPBT is generally
robust against the formation of various defects, we find that anti-site defect
formation of $\text{Mn}_{\text{Pb}}$ is the most likely to occur, with
corresponding energy less than 60~meV. This can be attributed to the
significant energy cost from compressive strain at the PbTe layer. Our findings
suggest that MPBT is on the brink of stability in terms of thermodynamics and
defect formation, underscoring the importance of conducting systematic analyses
of the stability of proposed TIs, including MPBT, for their practical
utilization. This study offers valuable insights into the design and synthesis
of desirable magnetic TI materials with robust stabilities.
