Phonon-induced dephasing of two-electron spin states in two vertically stacked self-assembled GaAs/InGaAs quantum dots is studied. A pure dephasing process due to elastic phonon scattering is found to dominate at low temperatures. This process is independent of the spin-orbit coupling and does not require the presence of a magnetic field. It relies on interdot tunneling and the Pauli principle, which make the double quantum dot gate for spin based quantum computing possible, and therefore cannot be avoided.