First, we note that $\cos 2000^{\circ} = \cos (2000^{\circ} - 360^{\circ} \times 5) = \cos 200^{\circ}$. Since $\cos 200^{\circ} = \cos (180^{\circ} + 20^{\circ}) = -\cos 20^{\circ}$, we need to find $\cos 20^{\circ}$ in terms of $a$.

We know that $\tan 20^{\circ} = a$, which means $\tan^2 20^{\circ} = a^2$. Using the identity $\tan^2 \theta + 1 = \sec^2 \theta$, we have $a^2 + 1 = \sec^2 20^{\circ}$, so $\sec 20^{\circ} = \sqrt{a^2 + 1}$. Since $\sec \theta = \frac{1}{\cos \theta}$, we have $\cos 20^{\circ} = \frac{1}{\sqrt{a^2 + 1}}$.

Therefore, $\cos 200^{\circ} = -\cos 20^{\circ} = -\frac{1}{\sqrt{a^2 + 1}}$.

The correct answer is $\boxed{\text{D}}$.