A flow-through folding membrane accumulator comprises a hollow cylindrical housing and a folding flexible membrane disposed in an interior volume of the housing. The membrane defines an interior liquid channel extending through the membrane along a longitudinal axis, and liquid coolant flows from an inlet of the accumulator through the interior liquid channel to an outlet of the accumulator. The membrane is sealed to the housing such that an air-tight cavity filled with a gas is formed between the housing and the membrane. The cavity radially surrounds the internal liquid channel, with the membrane separating the cavity and the internal liquid channel. Volumes of the cavity and channel change in response to deformation of the membrane and based on pressures of the liquid coolant. The membrane may comprise multiple folds extending parallel to the longitudinal axis and distributed circumferentially around the axis.