Most commonly the term has been used by authors such as John Armitage (‘The prospect of astro-palaeontology’) to describe the study of prehistoric extraterrestrial life. Astropalaeontology in this sense is a sub-discipline of astrobiology, and in fact as Armitage argued it may be more likely that we will discover the remains of extinct extraterrestrial life before we discover a living example. Exopalaeontology and astropalaeobiology are synonyms, and I’ve also seen palaeoastrobiology. Xenopalaeontology doesn’t seem to have caught on in serious circles.
It’s worth noting again that astropalaeontology is a separate discipline from xenoarchaeology, as terrestrial palaeontology is from archaeology (although in both cases the fields share a blurry boundary). Archaeological fields are more concerned with cultural remains, which are created by sophonts.
Secondly the term has been used to describe the study of stellar evolution (as an analogy to palaeontology’s study of the evolution of life). See the paper ‘The evolution of the mass-metallicity relation in SDSS galaxies uncovered by astropaleontology’ by Asari et al. You also see terms like stellar palaeontology, or galactic palaeontology used to discuss this kind of research.
Here’s a summary of ‘galactic palaeontology’ from the University of Sydney:
Ultra metal-poor stars are the living fossils of the stellar kingdom. Although elements heavier than Helium only make up a tiny fraction of any star, they have a profound effect on the stellar structure.
Consequently stars born when the universe was substantially younger, before heavy elements were formed, should stand out from the crowd exhibiting dramatically different physical and thermal structure — or so the theoretical models tell us. Because these fossil stars are rare and far from Earth, nobody has ever been able to examine one in detail. Until now.
If this stellar use of the term was more common, I’d suggest that astropalaeobiology is the better word to describe the study of extraterrestrial fossils. The biology part would avoid confusion, and it ties in with astrobiology, the most common term for the study of extraterrestrial life in use.
Thirdly, I’ve seen one use of the term astropalaeontology to describe study of the way the evolution of life on earth has been affected by astronomical events. See this 2009 article in National Geographic which discusses the theory that a gamma-ray burst might have caused a mass extinction.