  We report thermopower measurements under hydrostatic pressure on the cuprate
superconductor La$_{1.6-x}$Nd$_{0.4}$Sr$_x$CuO$_4$ (Nd-LSCO), at
low-temperature in the normal state accessed by suppressing superconductivity
with a magnetic field up to $H = 31$ T. Using a newly developed AC thermopower
measurement technique suitable for high pressure and high field, we track the
pressure evolution of the Seebeck coefficient $S$. At ambient pressure and low
temperature, $S/T$ was recently found to suddenly increase in Nd-LSCO at the
pseudogap critical doping $p^{\star} = 0.23$, consistent with a drop in carrier
density $n$ from $n = 1 + p$ above $p^{\star}$ to $n = p$ below. Under a
pressure of 2.0 GPa, we observe that this jump in $S/T$ is suppressed. This
confirms a previous pressure study based on electrical resistivity and Hall
effect which found $dp^{\star}/dP \simeq - 0.01$ holes/GPa, thereby reinforcing
the interpretation that this effect is driven by the pressure-induced shift of
the van Hove point. It implies that the pseudogap only exists when the Fermi
surface is hole-like, which puts strong constraints on theories of the
pseudogap phase. We also report thermopower measurements on Nd-LSCO and
La$_{1.8-x}$Eu$_{0.2}$Sr$_x$CuO$_4$ in the charge density-wave phase near $p
\sim 1/8$, which reveals a weakening of this phase under pressure.
