Here we report the transient cooling performance of a body vascularized with tree-shaped channels supplied with coolant, which flows from one side of the body to the other. The vasculature consists of trees that alternate with upside down trees. Heat is generated volumetrically through the body at t=0. A time delay td separates the start of the flow of coolant from the start of heating. Three-dimensional simulations of conduction and convection in the solid-fluid composite show the formation and evolution of hot spots in the material. If the delay is not short enough, the maximum temperature of the body overshoots the maximum allowable level. The paper shows how to design the cooling delay time such that the hot-spot temperature does not exceed the safe level represented by the maximum temperature in the limit of steady state operation. The critical delay time is determined as a function of the applied pressure difference and the complexity of the dendritic flow architecture. © 2009 American Institute of Physics.