Geologist here. Earth's thermal evolution is unsolved decades after we figured out the stars. The Sun's temperature adjusts to energy production and loss on a 10^6 yr timescale (scaling as the time for a photon to diffuse from the heart of the Sun to the cold surface): << the Sun's age. The Earth's temperature adjusts to internal energy production and surface losses on a billion year timescale: of order the Earth's age. Planets have long memories and history matters. The Sun is reasonably well-mixed. Surface spectroscopy probes the make-up of the whole star. Earth is less well-stirred. Seismic imaging of the deep earth maps the edges of vast pods of material, radioactivity unknown, composition unknown (but definitely distinct from the near-surface stuff), age unconstrained but plausibly as old as the planet [1]. Structure and composition matter [2]. It's a hard problem.

But it matters. When you look up at the night sky far from cities, for every star you see there's a habitable-zone Earth-radius planet that's closer [3]. We didn't know that six months ago. We think (for good, but circumstantial reasons) that complex life requires volcano-tectonic resurfacing - necessarily, a hot interior. Given that habitable-zone Earth-radius planets are not in short supply, the difference between fast and slow cooling for planets like Earth is the difference between a Galaxy where most every star system is habitable and one where almost all the planets are cinders.

The core-mantle boundary heat flux Q_CMB estimated in this paper constrains the mantle energy balance

d(E_mantle)/dt ~ Q_CMB - Q_surf + H_radioactive

Surface heat flux Q_surf is ~46 terawatts. Mantle radioactivity H_radioactive is not well constrained but about 10 terawatts [2]. The implication is that despite the high core heat flux, Earth's mantle is cooling fast - maybe 100 microkelvins per century. Volcanism will therefore shut down in much less time than the remaining main sequence lifetime of the Sun. Absent human intervention, the reddening of the Sun won't kill the biosphere, the Earth will.

As the mantle cools, the temperature contrast between the mantle and the core will no longer sustain core convection. Then Earth's magnetic field will power down. Without geo-dynamo shielding against galactic and solar radiation, bad things may happen: the rapid shutdown of Mars' dynamo is one hypothesis for the deterioration of Mars climate ~4 Gyr ago [4]. On the other hand, Earth's magnetic field strength decreased by a factor of 20 during the Laschamp Event ~41000 years ago [5], with no known effects on biology (or human culture).

Diamond-anvil experiments are tough; few grad students make it past quals without breaking a diamond or two. The diamond-anvil technique is hitting diminishing returns, so modest advances are (rightly) celebrated. The same is true for deep-earth seismology and mantle geochemistry. A good new method is mapping the antineutrino flux from Earth. Antineutrinos are produced by radioactive decay and move in a straight line from source to surface. Mapping the Earth with geoneutrino observatories in the deep sea would help determine the power source for plate tectonics [6].

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[1] Garnero & McNamara: http://mcnamara.asu.edu/Publications/pdfs/Garnero_and_McNama...

[2] Korenaga, "Urey Ratio and The Structure and Evolution of Earth's Mantle", http://people.earth.yale.edu/sites/default/files/korenaga08d...
Korenaga is the best mid-career theorist actively working on this problem.

[3] New result, from several teams working independently to analyze the Kepler dataset:
Caltech http://arxiv.org/abs/1303.3013 (read this one first); Harvard team #1 http://arxiv.org/abs/1302.1647; Harvard team #2 http://arxiv.org/abs/1301.0842; Berkeley http://arxiv.org/abs/1304.0460. I'm assuming 0.1 stars per cubic parsec.

[4] Lillis et al: http://seismo.berkeley.edu/~manga/lillisetal2008b.pdf. Later work broadly supports his conclusion that the dynamo died fast and early in Mars history. SETI Institute talk: https://www.youtube.com/watch?v=REiKzxWbzrQ
It is not known whether loss of the magnetic field had a big or small effect on the Great Drying of Mars. Measuring modern atmosphere/water loss rates from modern Mars is the goal of the MAVEN mission, which launches this Nov - http://lasp.colorado.edu/home/maven/

[5] Known from ice-core spikes in beryllium-10 (isotope produced by cosmic radiation hitting Earth's atmosphere) as well as magnetic paleo-intensity measurements in sediments.

[6] http://www.phys.hawaii.edu/~sdye/hanohano.html. A knuckle is that SSBN reactors also emit neutrinos and neutrinos cannot be shielded, so deep-sea geoneutrino detectors could be strategically destabilizing. In practice either angular resolution or massive size would be needed to make deep-sea neutrino detectors useful to militaries.