What does geochemical data tell us about Earth's earliest palaeoclimatic conditions? Given that the age of the oldest zircon samples are about 4.4 billion years old, and meteorites and other extraterrestrial samples can also give us some indication about the Earth's composition into the Hadean era (source: Zircon Chronology: Dating the Oldest Material on Earth.

Many resources refer to the earliest atmosphere as being (for example):


  As a result of the high temperatures at the center of the Earth, and due to volcanic activity, the crust emitted halogen gasses, ammonia, hydrogen, carbon dioxide, methane, water vapor, and other gasses. In the following 100 million years, these gasses accumulated to form the primordial atmosphere.  This atmosphere was quite similar to the atmosphere of Titan, one of the larger moons of Saturn.  The primordial atmosphere is believed to have reached a pressure of 250 atmospheres and would have been extremely toxic to life as we now know it


Source: The Hadean Eon

Many mechanisms are suggested in this and many other resources, from impacts to volcanic outgassing etc. But what does measured geochemical data tell us about the Earth's earliest palaeoclimatic conditions?

This question is related to the previous questions:


What, if any, paleoclimate data can be derived from igneous rocks?
What, if any, paleoclimate data can be derived from stromatolite fossils?


But, this question is looking at the earliest palaeoclimatic conditions of the Earth.
The first thing to note here is that the zircons you mention in your question are the only thing we have dating from the Hadean: no known Hadean continental or oceanic crust have been found to date. So, all informations come from these zircons.

What do they tell us? 

The rare earth element (REE) composition of these zircons (and in particular the zoning of said REE inside the mineral) suggests that they formed in a magma derived from the melting of a continental crust (Wilde et al. 2001). Their high $\delta^{18}\mathrm{O}$ composition also indicates that the magma was at some point in contact with liquid water (also Wilde et al. 2001): both continents and oceans were present in the Hadean. The presence of liquid water already implies cooler temperature than what was previously thought (Valley et al. 2002).

With regards to climate per say, the Lithium composition (high concentration, light isotopes) of the zircons (Ushikobo et al. 2008) have been interpreted as evidence of an intense chemical continental weathering. This weathering could have several explanations but one possibility retained by Ushikobo et al. 2008 (see also Sleep 2010) is a CO2-rich atmosphere with surface temperatures above 50°C.


Sleep, 2010. The Hadean-Archaean Environment. Cold Spring Harbor Perspectives in Biology, 2: a002527.
Ushikobo, Kita, Cavosie, Wilde, Rudnick &amp; Valley, 2008. Lithium in Jack Hills zircons: Evidence for extensive weathering of Earth's earliest crust. Earth and Planetary Science Letters, 272: 666-676.
Valley, Peck, King &amp; Wilde, 2002. A cool early Earth. Geology, 30: 351-354.
Wilde, Valley, Peck &amp; Graham, 2001. Evidence from detrital zircons for the
existence of continental crust and oceans on the Earth 4.4 Gyr ago. Nature, 409: 175-178.