When initial concentration of the reactant is doubled, the half-life period of a zero order reaction:

For a zero-order reaction, the half-life is given by:
$$t_{1/2}=\frac{[A{]}_0}{2k}$$
Since $t_{1/2}$ is directly proportional to the initial concentration $[A{]}_0$, doubling $[A{]}_0$ will double the half-life period.