An electron beam accompanying a heavy particle beam in a straight can serve as a thermostat for circulating ion beam. In conventional low energy electron coolers, the electron beam is immersed in a continuous solenoid, which provides a tightly confined and calm beam in cooling
section. The continuity of solenoid is hardly realizable at relativistic energies. A possibility is proved conceptually to use an extended solenoid in the gun and the cooling section only, applying lumped focusing for the rest of electron transport line. Such solution being a non-trivial idea as an optical concept for electron cooler allows to incorporate with RF linacs to obtain high quality e-beams with no limitation-in principle on maximum energy of electrons. The breakthrough can be enhanced by use of energy-recovered linacs and electron recirculators as cooler rings.  Proposed optimum focusing for ion beam in cooling section will be discussed. Conceptual experience and design stages of Medium and High Energy coolers for hadron colliders will be
illustrated.