Electron and stochastic cooling are proven methods for cooling low-energy hadron beams, but at present there is no way of cooling hadrons as they near the TeV scale. In the 1980s, Derbenev suggested that electron instabilities, such as free-electron lasers, could create collective space charge fields strong enough to correct the hadron energies. This Letter presents a variation on Derbenev's electron cooling scheme using the microbunching instability as the amplifier. The large bandwidth of the instability allows for faster cooling of high-density beams. A simple analytical model illustrates the cooling mechanism, and simulations show cooling rates for realistic parameters of the Large Hadron Collider.