  The ideal exoplanets to search for life are those within a star's habitable
zone. However, even within the habitable zone planets can still develop
uninhabitable climate states. Sustaining a temperate climate over geologic
($\sim$Gyr) timescales requires a planet contain sufficient internal energy to
power a planetary-scale carbon cycle. A major component of a rocky planet's
energy budget is the heat produced by the decay of radioactive elements,
especially $^{40}$K, $^{232}$Th, $^{235}$U and $^{238}$U. As the planet ages
and these elements decay, this radiogenic energy source dwindles. Here we
estimate the probability distribution of the amount of these heat producing
elements (HPEs) that enter into rocky exoplanets through Galactic history, by
combining the system-to-system variation seen in stellar abundance data with
the results from Galactic chemical evolution models. Using these distributions,
we perform Monte-Carlo thermal evolution models that maximize the mantle
cooling rate. This allows us to create a pessimistic estimate of lifetime a
rocky, stagnant-lid exoplanet can support a global carbon cycle and temperate
climate as a function of its mass and when it in Galactic history. We apply
this framework to a sample of 17 likely rocky exoplanets with measured ages, 7
of which we predict are likely to be actively degassing today despite our
pessimistic assumptions. For the remaining planets, including those orbiting
TRAPPIST-1, we cannot confidently assume they currently contain sufficient
internal heat to support mantle degassing at a rate sufficient to sustain a
global carbon cycle or temperate climate without additional tidal heating or
undergoing plate tectonics.
