We further develop the gravitational model, Thomas-Whitehead Gravity (TW Gravity), that arises when projective connections become dynamical fields. TW Gravity has its origins in geometric actions from string theory where the TW projective connection appears as a rank two tensor, $mathcal{D}_{ab}$, on the spacetime manifold. Using a Gauss-Bonnet (GB) action built from the $(mathrm{d}+1)$-dimensional TW connection, and applying the tensor decomposition $mathcal{D}_{ab} = D_{ab} + 4Lambda /(mathrm{d}(mathrm{d}-1)) g_{ab}$, we arrive at a gravitational model made up of a $mathrm{d}$-dimensional Einstein-Hilbert + GB action sourced by $D_{ab}$ and with cosmological constant $Lambda$. The $mathrm{d}=4$ action is studied and we find that $Lambda propto 1/J_0$, with $J_0$ the coupling constant for $D_{ab}$. For $Lambda$ equal to the current measured value, $J_0$ is on the order of the measured angular momentum of the observable Universe. We view this as $Lambda$ controlling the scale of patches of the Universe that acquire angular momentum, with the net angular momentum of multiple patches vanishing, as required by the cosmological principle. We further find a universal axial scalar coupling to all fermions where the trace, $mathcal{D} = mathcal{D}_{ab}g^{ab}$ acts as the scalar. This suggests that $mathcal{D}$ is also a dark matter portal for non-standard model fermions.
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