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We consider cosmological aspects of the Dark Dimension (a mesoscopic dimension of micron scale), which has recently been proposed as the unique corner of the quantum gravity landscape consistent with both the Swampland criteria and observations. In particular we show how this leads, by the universal coupling of the Standard Model sector to bulk gravitons, to massive spin 2 KK excitations of the graviton in the dark dimension (the "dark gravitons") as an unavoidable dark matter candidate. Assuming a lifetime for the current de Sitter phase of our universe of order Hubble, which follows from both the dS Swampland Conjecture and TCC, we show that generic features of the dark dimenson cosmology can naturally lead to the correct dark matter density and a resolution of the cosmological coincidence problem, where the matter/radiation equality temperature ($T\sim$ 1 eV) coincides with the temperature where the dark energy begins to dominate. Thus one does not need to appeal to Weinberg's anthropic argument to explain this coincidence. The dark gravitons are produced at $T\sim$ 4 GeV, and their composition changes as they mainly decay to lighter gravitons, without losing much total mass density. The mass of dark gravitons is $m_{\text{DM}}\sim 1-100$ keV today.