We study the Kardar-Parisi-Zhang (KPZ) growth equation in one dimension with a noise variance $c(t)$ depending on time. We find that for $c(t)\propto t^{-α}$ there is a transition at $α=1/2$. When $α>1/2$, the solution saturates at large times towards a non-universal limiting distribution. When $α<1/2$ the fluctuation field is governed by scaling exponents depending on $α$ and the limiting statistics are similar to the case when $c(t)$ is constant. We investigate this problem using different methods: (1) Elementary changes of variables mapping the time dependent case to variants of the KPZ equation with constant variance of the noise but in a deformed potential (2) An exactly solvable discretization, the log-gamma polymer model (3) Numerical simulations.