No, energy is not always conserved in general relativity. There is no way to define the energy of an isolated system as a function of the state of the system such that total energy and momentum is conserved when ever 2 systems combine or undergo a hyperbolic orbit and the definition simplifies to the definition in special relativity for low mass and density. An electromagnetic field can't affect a gravitational field directly but can accelerate a particle which in turn affect the gravitational field and in the absense of matter, an electromagnetic field can't affect a gravitational field. An electric field can't accelerate a charged black hole because there's no matter outside its event horizon for it to accelerate and in turn affect the fabric of space outside the event horizon so energy is not always conserved in general relativity.

No, energy is not always conserved in general relativity. There is no way to define the energy of an isolated system as a function of the state of the system such that total energy and momentum is conserved when ever 2 systems combine or undergo a hyperbolic orbit and the definition simplifies to the definition in special relativity for low mass and density. An electromagnetic field can't affect a gravitational field directly but can accelerate a particle which in turn affect the gravitational field and in the absense of matter, an electromagnetic field can't affect a gravitational field. According to the Wikipedia article No hair theorem, the observable state of a black hole is completely described by its mass, charge, and angular momentum. An electric field can't accelerate a charged black hole because there's no matter outside its event horizon for it to accelerate and in turn affect the fabric of space outside the event horizon so energy is not always conserved in general relativity.

This answer is not completely my own and uses some ideas from Lubos Motl's answer at Is the law of conservation of energy still valid?. Energy is not conserved in general relativity. Time symmetry has been show to be equivalent to the conservation of energy in general relativity.

  There is no way to define the energy of an isolated system as a function of the state of the system such that total energy and momentum is conserved when ever 2 systems combine or undergo a hyperbolic orbit and the definition simplifies to the definition in special relativity for low mass and density. An electromagnetic field can't affect a gravitational field directly but can accelerate a particle which in turn affect the gravitational field and in the absense of matter, an electromagnetic field can't affect a gravitational field. An electric field can't accelerate a charged black hole because there's no matter outside its event horizon for it to accelerate and in turn affect the fabric of space outside the event horizon so energy is not always conserved in general relativity. Since energy is no conserved, time must not be symmetric either. Indeed, I can show that it's not symmetric. Although the fundamental laws are time symmetric everywhere except at the singularity of a black hole, not everything that can happen according to the big bang theory can happen in reverse, a binary star system never absorbs an incoming gravitational wave spiraling out in the process. Also, matter can enter a black hole but can never leave a black hole and a black hole can never uncollapse into matter.

No, energy is not always conserved in general relativity. There is no way to define the energy of an isolated system as a function of the state of the system such that total energy and momentum is conserved when ever 2 systems combine or undergo a hyperbolic orbit and the definition simplifies to the definition in special relativity for low mass and density. An electromagnetic field can't affect a gravitational field directly but can accelerate a particle which in turn affect the gravitational field and in the absense of matter, an electromagnetic field can't affect a gravitational field. An electric field can't accelerate a charged black hole because there's no matter outside its event horizon for it to accelerate and in turn affect the fabric of space outside the event horizon so energy is not always conserved in general relativity.