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.
Categories: gr-qc cond-mat.stat-mech hep-th