  Venus may have had both an Earth-like climate as well as extensive water
oceans and active (or incipient) plate tectonics for an extended interval of
its history. The topographical power spectrum of Venus provides important clues
to the planet's past evolution. By drawing detailed contrast with the strong
low-order odd-$l$ dominated global topography of Earth, we demonstrate that the
relatively flat Venusian topography can be interpreted to have arisen from the
transition from active terrestrial-like plate tectonics to the current stagnant
lid configuration at a time $\tau = 544^{+886}_{-193}$ million years before
present. This scenario is plausible if loss of oceans and the attendant
transition to a CO$_2$-dominated atmosphere were accompanied by rapid
continental-scale erosion, followed by gradual lava resurfacing at an outflow
rate $\sim$ 1 km$^{3}$ yr$^{-1}$. We study Venus' proposed topographical
relaxation with a global diffusion-like model that adopts terrestrial erosion
rates scaled to account for the increased rainfall and temperatures that would
accompany a planet-wide transition from an Earth-like climate to the runaway
greenhouse climate that could ultimately yield present-day Venus, with an
estimate of $5.1^{+1.8}_{-1.1}$ Myr if the global erosion operated as
efficiently as that of a typical bedrock river basin on Earth.
