To find the work done by an ideal gas along the path $ABCD$ on the given $P-V$ diagram, we calculate the work done on each segment individually and then sum them up.

Segment $AB$: The work is given by $\text{w}_{AB} = P_0 \times V_0$, contributing $P_0 V_0$.

Segment $BC$: No volume change occurs along this path, resulting in zero work: $\text{w}_{BC} = 0$.

Segment $CD$: The gas compresses, performing negative work: $\text{w}_{CD} = -(2P_0 \times 2V_0) = -4P_0 V_0$.

Adding the work done in each segment:

$ \begin{align*} \text{w}_{ABCD} &= \text{w}_{AB} + \text{w}_{BC} + \text{w}_{CD} \\ &= P_0 V_0 + 0 - 4P_0 V_0 \\ &= -3P_0 V_0 \end{align*} $

Thus, the total work done by the gas along the path $ABCD$ is $-3P_0 V_0$.