  The newly discovered BaPt$_2$As$_2$ shows a structural distortion at around
275~K, followed by the emergence of superconductivity at lower temperatures.
Here we identify the presence of charge density wave (CDW) order at room
temperature and ambient pressure using single crystal x-ray diffraction, with
both a superlattice and an incommensurate modulation, where there is a change
of the superlattice structure below $\simeq$ 275~K. Upon applying pressure,
BaPt$_2$As$_2$ shows a rich temperature-pressure phase diagram with multiple
pressure-induced transitions at high temperatures, the emergence or
disappearance of which are correlated with sudden changes in the
superconducting transition temperature $T_c$. These findings demonstrate that
BaPt$_2$As$_2$ is a promising new system for studying competing interactions
and the relationship between high-temperature electronic instabilities and
superconductivity.
