We have investigated the influence of hydrostatic pressure on the tunneling processes between two independently contacted two‐dimensional electron gas systems which are separated by a doped AlGaAs barrier. For magnetotunneling data, the pressure dependence of the GaAs effective massm* was determined betweenp=0 and 10 kbar. In this range, we find a linear pressure dependence ofd(m*/me)/dp=8×10−4kbar−1for the electrons in the two‐dimensional channels. At higher pressures several new resonances are observed, which correspond to transition energies in the order of 4 meV. These effects are most probably explained by local phonon modes related to theDXcenters inside the barrier.