By employing Tsallis' extensive but non-additive delta-entropy, we formulate the first two laws of thermodynamics for gravitating systems. By invoking Carath & eacute;odory's principle, we pay particular attention to the integrating factor for the heat one-form. We show that the latter factorizes into the product of thermal and entropic parts, where the entropic part cannot be reduced to a constant, as is the case in conventional thermodynamics, due to the non-additive nature of S-delta. The ensuing two laws of thermodynamics imply a Tsallis cosmology, which is then applied to a radiation-dominated universe to address the Big Bang nucleosynthesis and the relic abundance of cold dark matter particles. It is demonstrated that the Tsallis cosmology with the scaling exponent delta similar to 1.499 (or equivalently, the anomalous dimension Delta similar to 0.0013) consistently describes both the abundance of cold dark matter particles and the formation of primordial light elements, such as deuterium H-2 and helium He-4. Salient issues, including the zeroth law of thermodynamics for the delta-entropy and the lithium Li-7 problem, are also briefly discussed.
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