The gravitational field equations were derived in general relativity (GR) using the assumption of their covariance relative to arbitrary transformations of coordinates. Over the past century, it has been repeatedly expressed as an opinion that such equality of all coordinate systems may not correspond to reality. In this article, general covariance is restricted by the introduction of a constraint. This constraint is interpreted from a physical point of view as a sufficient condition for the adiabaticity of the process of the evolution of the space–time metric. The original equations of the theory of gravity with the constraint are formulated. On this basis, a unified model of the evolution of the modern, early, and very early Universe is constructed that is consistent with the observational astronomical data but does not require the hypotheses of the existence of dark energy, dark matter or inflatons. It is claimed that: the gravitational field is the main source of energy of the Universe, the maximum global energy density in the Universe was 64 orders of magnitude smaller the Planckian one, and the entropy density is 18 orders of magnitude higher the value predicted by GR. The value of the relative density of neutrinos at the present time and the maximum temperature of matter in the early Universe are calculated. The wave equation of the gravitational field is formulated, its solution is found, and the wave function of the very early Universe is constructed. It is shown that the birth of the Universe was random.