Note that this has nothing to do with energy conservation per se. However, relativity serves up a new twist. In Newtonian physics, energy conservation and momentum conservation are two separate laws. Very massive objects emitting light Light from the Sun appears redshifted to an Earthbound astronomer. In quasi-Newtonian terms, we might say that light loses kinetic energy as it climbs out of the gravitational well of the Sun, but gains potential energy.
General relativity looks at it differently.
The Schwarzschild metric is both static and asymptotically flat, and energy conservation holds without major pitfalls. For further details, consult MTW, chapter 25. So its orbital period should change. Hulse and Taylor were awarded the Nobel prize in 1993. The need to understand this situation better has kept GR theoreticians busy over the last few years.