  Based on the equivalence of A-stability and G-stability, the energy technique
of the six-step BDF method for the heat equation has been discussed in
[Akrivis, Chen, Yu, Zhou, Math. Comp., Revised]. Unfortunately, this theory is
hard to extend the time-fractional PDEs. In this work, we consider three types
of subdiffusion models, namely single-term, multi-term and distributed order
fractional diffusion equations. We present a novel and concise stability
analysis of time stepping schemes generated by $k$-step backward difference
formula (BDF$k$), for approximately solving the subdiffusion equation. The
analysis mainly relies on the energy technique by applying Grenander-Szeg\"{o}
theorem. This kind of argument has been widely used to confirm the stability of
various $A$-stable schemes (e.g., $k=1,2$). However, it is not an easy task for
the higher-order BDF methods, due to the loss the $A$-stability. The core
object of this paper is to fill in this gap.
