  In this paper we are concerned with contact process with random recovery
rates on open clusters of bond percolation on $\mathbb{Z}^d$. Let $\xi$ be a
positive random variable, then we assigned i. i. d. copies of $\xi$ on the
vertices as the random recovery rates. Assuming that each edge is open with
probability $p$ and $\log d$ vertices are occupied at $t=0$, we prove that the
following phase transition occurs. When the infection rate
$\lambda<\lambda_c=1/({p{\rm E}\frac{1}{\xi}})$, then the process dies out at
time $O(\log d)$ with high probability as $d\rightarrow+\infty$, while when
$\lambda>\lambda_c$, the process survives with high probability.
