  We present a new stability and error analysis of fully discrete approximation
schemes for the transient Stokes equation. For the spatial discretization, we
consider a wide class of Galerkin finite element methods which includes both
inf-sup stable spaces and symmetric pressure stabilized formulations. We extend
the results from Burman and Fern\'andez [\textit{SIAM J. Numer. Anal.}, 47
(2009), pp. 409-439] and provide a unified theoretical analysis of backward
difference formulae (BDF methods) of order 1 to 6. The main novelty of our
approach lies in the use of Dahlquist's G-stability concept together with
multiplier techniques introduced by Nevannlina-Odeh and recently by Akrivis et
al. [\textit{SIAM J. Numer. Anal.}, 59 (2021), pp. 2449-2472] to derive optimal
stability and error estimates for both the velocity and the pressure. When
combined with a method dependent Ritz projection for the initial data,
unconditional stability can be shown while for arbitrary interpolation,
pressure stability is subordinate to the fulfillment of a mild inverse CFL-type
condition between space and time discretizations.
