We consider the stability and convergence analysis of pressure stabilized finite element approximations of the transient Stokes' equation. The analysis is valid for a class of symmetric pressure stabilization operators. Provided the initial data is chosen as a specific (pressure stabilization dependent) Ritz-projection, we get unconditional stability and optimal convergence for both pressure and velocity approximations, in natural norms. For arbitrary interpolations of the initial data, a condition between the space and time discretization parameters has to be verified in order guarantee pressure stability