general relativity - Energy-momentum tensor for dust

We all know that the energy-momentum tensor for dust is just $T^{\alpha\beta}=\rho_0v^\alpha v^\beta,$ where $\rho_0$ is the mass density in the dust's rest frame and $v^α$ is the dust's four-velocity. I'm trying to derive the dust energy momentum tensor from the equation $T_{αβ}=-\frac{2}{\sqrt{-g}}\frac{δS_M}{δg_{αβ}}$ but I'm getting the wrong answer.

The action for dust is

$$S=\int -\rho_0\sqrt{-g}d^4x.$$

Thus

$$\frac{\delta S}{\delta g^{\alpha\beta}}=\frac{\delta (-\rho_0)}{\delta g^{\alpha\beta}}\sqrt{-g}-\rho_0\frac{\delta \sqrt{-g}}{\delta g^{\alpha\beta}}.$$

To evaluate $\frac{\delta (-\rho_0)}{\delta g^{\alpha\beta}}$, I define $K_\alpha=\rho_0 v_\alpha$. Then $\rho_0=\sqrt{g^{\alpha\beta}K_\alpha K_\beta}$ and thus $\delta \rho_0=\frac{1}{2\rho_0}K_\alpha K_\beta \delta g^{\alpha \beta}.$ It follows that $$T_{\alpha\beta}=\rho_0 v_\alpha v_\beta-\rho_0 g_{\alpha\beta}.$$

There's an extra term that I can't get rid of. Any idea where I went wrong?