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We prove the spectral instability of the complex cubic oscillator$-\frac{d^2}{dx^2}+ix^3+i\alpha x$ for non-negative values of the parameter$\alpha$, by getting the exponential growth rate of $\|\Pi_n(\alpha)\|$, where$\Pi_n(\alpha)$ is the spectral projection associated with the $n$-theigenvalue of the operator. More precisely, we show that for all non-negative$\alpha$ \[  \lim\limits_{n\to+\infty}\frac{1}{n}\log\|\Pi_n(\alpha)\| =\frac{\pi}{\sqrt{3}}. \]

Spectral projections of the complex cubic oscillator