  By employing Tsallis' extensive but non-additive $\delta$-entropy, we
formulate the first two laws of thermodynamics for gravitating systems. By
invoking Carath\'{e}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$$\sim$$1.499$ (or equivalently, the anomalous
dimension $\Delta\sim0.0013$) consistently describes both the abundance of cold
dark matter particles and the formation of primordial light elements, such as
deuterium ${}^{2}\!H$ and helium ${}^{4}\!He$. Salient issues, including the
zeroth law of thermodynamics for the $\delta$-entropy and the lithium
${}^{7}\!Li$ problem, are also briefly discussed.
