Most experimental research in matter under extreme pressures is concerned with recreating conditions within the interiors of planets and stars (the latter falls under the field of high energy density physics). The temperatures involved therefore tend to be very high. However, there's no inherent conflict between high pressures and low temperatures, it's just that temperature tends to increase when you compress something. Compress an ideal gas, for example, and it will heat up. Let it sit in its compressed state for a while though, and it will cool back down despite remaining under high pressure.

This is true for solids and liquids too (putting any phase transitions aside), though they are much less compressible. The core of the Earth will eventually cool too, though it's currently kept at high temperature by the radioactive decay of heavy elements. Diamond anvil cells, however, can reach pressures exceeding those at the center of the earth in a laboratory setting, and some DACs can even be cooled to cryogenic temperatures. This figure on Wikipedia suggests cryo-DACs can be used to reach pressures up to 350 GPa at cryogenic temperatures. As an example, a quick search turns up a paper (arxiv version) that makes use of a DAC to study media at liquid nitrogen temperatures and pressures up to 10 GPa (~3% the pressure at the center of the Earth). Search around and I'm sure you can find others.