lpetrich

Di Giulio M.
The universal ancestor was a thermophile or a hyperthermophile: tests and further evidence.
J Theor Biol 2003 Apr 7;221(3):425-36

The method used was a combination of:

Reconstruction of several ancestral protein sequences from major groups of organisms ("Archean-Park proteins"):

Bacteria / Eubacteria
Archaea / Archaebacteria
Eukarya
Their shared ancestor

Working out a "thermophily index", a translation of a protein's amino-acid content into a likely optimum temperature for that protein's functioning.

And the results are:

Eukarya's ancestor was mesophilic, preferring temperatures ~ 10 - 40 C

The ancestors of Bacteria, Archaea, and of all three are (hyper)thermophilic, preferring temperatures ~ 40 - 100+ C

This is consistent with the hypothesis of the origin of life being in oceanic hot springs, with eukaryotic organisms coming after the original prokaryotic ones.