$\gamma$. Hence we define $$\mu^+ = \gamma \mu^- + \chi  \qquad \mu^+ + \mu^- =\mu,$$ where $\chi$ is a constant. Using nonsymmetric parametrization, the rate of complexity growth is given by $$\label{deriv}
\dot{\mathcal{C}}=\frac{1}{\pi}\left(\frac{d I}{d \mu}\right) = \frac{1}{\pi}\left(\frac{\gamma}{1+\gamma}\frac{d I}{d \mu^+}  + \frac{1}{1+\gamma}\frac{d I}{d \mu^-}\right).$$

A1 Initial time {#a1-initial-time .unnumbered}
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In this case the GHY terms near the singularities change as $$I_{GHY}^{sing}=\frac{\Omega_{d-1} ~d~ \omega^{d-2}}{8 \pi G_{N}}\left( \mu^{-} - \mu^+\right).$$ Using the fact that $r_P=r_Q$ and also and the null joint terms become $$I_{J}^{null} = -\frac{\Omega_{d-1} ~(r_P )^{d-1}}{4 \pi G_{N}} \log \frac{\mathfrak{a}^2}{|f(r_P)|}.$$ Using the definition of the tortoise coordinate , one can find $$\frac{d r_P}{d \mu^+}= \frac{-f(r_P)}{2}, \qquad \frac{d r_P}{d \mu^-}= \frac{f(r_P)}{2}.$$ Hence we find the rate of complexity growth for nonsymmetric boundary times as $$\begin{aligned}
\dot{\mathcal{C}} =\frac{1}{\pi} ~\frac{d I}{d \mu}=\frac{1-\gamma}{1+\gamma} ~\frac{1}{\pi} ~ \left(4 M+ \frac{\Omega_{d-1} ~(d-1)~r_P^{d-2}}{8 \pi G_{N}} f(r_P) ~\log \frac{\mathfrak{a}^2}{f(r_P)}\right).\end{aligned}$$ It is clear that for the symmetric case, i.e., $\gamma=1$, the growth rate is zero.

A2 Late time {#a2-late-time .unnumbered}
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Now the boundary term near the future singularity results in $$\label{GHYlateap}
I_{GHY}^{sing}=\frac{\Omega_{d-1} ~d~ \omega^{d-2}}{8 \pi G_{N}}( \mu^-).$$ The contribution of the joint terms at $P$ and $Q$ is similar to the initial time where we have $$I_{J}^{null} = -\frac{\Omega_{d-1} 