I think in this case, the Oxygen-18 stable isotopes win in the
palaeotemperature proxy methods competition, e.g. Ice core records, linked
to global ice volume. They are relatively quick and cheap to run and the
cores give a stratigraphic framework to the results.
An important field and a specific example would be carbonate geochemistry.
The relative enrichment (or not) of the oxygen-18 stable isotopes in
secondary calcium carbonate is directly related to the temperature of the
water that the CaCO3 precipitated from. The equation for deriving the
temperature currently eludes me...
With AAR, the site specific nature of the reaction probably precludes an
accurate inference of palaeotemperature. It may be possible but it's
probably too inefficient to be a major proxy.
Amazing what you can do with some expensive machinery and a bit of old stuff
found in the ground! :)