In this paper, an adjoint sensitivity analysis with an iterative first-order gradient descent optimization technique is proposed in conjunction with the finite-difference time-domain (FDTD) method to design a broadband multi-layer planar electromagnetic wave absorber made of expanded polystyrene mixed with graphite powder. This analysis needs an additional adjoint field simulation for the sensitivity calculation of design parameters (thickness and density of absorbing materials), which can be computed by the FDTD technique with the backward time scheme. The result obtained from the proposed design method is compared with the point frequency matching method. A smaller thickness, carbon content, and better absorption performance of a multi-layer planar absorber are obtained over the frequency range of 0.4-5 GHz.