How do Christians understand the existence of God in relation to the scientific concept of energy?

God is pure Spirit if you had a spiritual scale, He would fill it.

St. Thomas Aquinas addresses this idea quite a bit.

This error seems to have originated from two statements of the ancients. For those who first began to observe the nature of things, being unable to rise above their imagination, supposed that nothing but bodies existed. Therefore they said that God was a body, which they considered to be the principle of other bodies. And since they held that the soul was of the same nature as that body which they regarded as the first principle, as is stated De Anima i, 2, it followed that the soul was of the nature of God Himself. According to this supposition, also, the Manichaeans, thinking that God was corporeal light, held that the soul was part of that light bound up with the body.

http://www.newadvent.org/summa/1090.htm

Like other answers state, your premise is that nothing else can exist. But consider those people whom you admire and look up to:

• Is there something about them that makes you go towards them?
• Does their reasoning impress you?
• Do they have a knack for rationality?

Consider what you feel to be excellent qualities in other human beings and then consider where their source is. What is the source of our rationality and moreover our desire to reason? It's not merely self-preservation.

I believe it's our soul and if we have a soul which animates our rationality, we must have a Creator who animates our soul.

No. God existed "before" the universe, time, and space and is entirely beyond the laws of physics. If you accept multiple universes, he is beyond all of them as well. He created the laws of physics. This is professed in most major Christian Creeds in some form, e.g., "I believe in God the Father, Creator of Heaven and Earth, of all that is seen and unseen." (emphasis mine), and is generally accepted by Christians. There are some exceptions, but all Catholics, Orthodox, and to the best of my knowledge all Protestants accept this as truth.

Energy can be defined in physics as the ability to perform work, as you say, and measured in joules or equivalent units. You ask whether we can ascribe energy to God in this sense. The ability of God to act in the world is a different question - I think you are asking about being able to measure God's own energy in some sense, whether this is kinetic energy or potential energy or anything else.

To calculate such a quantity, we would need to take measurements of some kind and somehow end up with a value in units of kg.m².s^-2 = J. This effort is stymied since there is no way to measure God's mass, length, etc., as these kind of measurements cannot apply to God. That is, we cannot measure God's energy for the same reason that we cannot measure his size - there is nothing special about energy in this regard. Can we measure his kinetic energy (= energy that depends on mass and velocity)? No, because God has neither. Potential energy (= energy that depends on position within a gravitational field)? No, because God is not located in a gravitational field, and indeed is not located at any point in space. Moreover, there is no particular reason to believe that even if he had energy, he would be subject to conservation of energy (we're not going to get very far trying to apply Noether's theorem).

This goes beyond a mere practical difficulty (I mean, it is not true to say that God has mass but we just can't calculate it). Theologically, attributes like "mass" cannot be applied to God, because God is an altogether different kind of thing than ordinary people or objects. The argument goes something like this. God's existence is pure (in scholastic terminology, he is actus purus, pure action) because his existence is (1) not dependent on anything else, (2) unchanging, and (3) he has no potential to be different from what he is. Hence it is a category error to apply any contingent attribute to God - any quality or quantity that we could imagine varying over time or in different circumstances. Mass is this kind of quantity. See Aquinas, Summa Theologica, Part 1 Q3 for the classic exposition. He talks about height, depth, colour and temperature being inapplicable concepts but it's the same idea.

Admittedly, some people have tried to apply concepts like "energy" to God. Teilhard de Chardin continually used the language of physics, in a dubious way, and in particular talked about humans having "spiritual energy" which would increase and merge with God's spiritual energy. But:

...he uses in metaphor words like energy, tension, force, impetus and dimension as if they retained the weight and thrust of their specific scientific usages. Consciousness, for example, is a matter upon which Teilhard has been said to have illuminating views. For the most part consciousness is treated as a manifestation of energy, though this does not help us very much because the word 'energy' is itself debauched; but elsewhere we learn that consciousness is a dimension, or is something with mass, or is something corpuscular and particulate which can exist in various degrees of concentration, being sometimes infinitely diffuse.

That's from the famous critical review by Peter Medawar of The Phenomenon of Man. It is a great reminder to be very careful about overinterpreting or misapplying concepts from physics.

Exodus 3:2-14 (NASB) says this:

The angel of the LORD appeared to [Moses] in a blazing fire from the midst of a bush; and he looked, and behold, the bush was burning with fire, yet the bush was not consumed... God called to him from the midst of the bush and said... the place on which you are standing is holy ground... God said to Moses, “I AM WHO I AM”; and He said, “Thus you shall say to the sons of Israel, ‘I AM has sent me to you.’”

Fire is normally the result of a release of energy in the form of light and heat caused by a chemical change in something that is burning. In normal burning, there is a change in state but the total amount of energy / matter in the universe remains constant.

In this passage, however, Moses witnesses something strange. He sees flames that exists without any visible means of support, ie. a release of energy without a corresponding change in the chemistry of the material universe. This violates his expectations, so naturally he wants a closer look. In doing so he perceives that God is present. According to Moses understanding, the bush is not the source of the energy being released, but rather God is.

This is consistent with the description in Genesis (a book often associated with Moses) which describes God as the source of all energy and matter in the universe. God needs nothing outside Himself to sustain Him, but rather is the source of everything else.

In the Bible, then, we see that God has the capacity to do work (ie. influence the universe) not because the concept of energy applies to Him, but because He is able to apply the concept of energy to the universe.

The theology of the Church of Jesus Christ of Latter-day Saints (Mormons) differs substantially from the other answers.

The differences start at the very beginning. The LDS faith teaches that creation ex nihilo is a false concept, and that God instead organized the world out of matter that was "without form" (Gen 1:2, more info and discussion here). This is understood to be matter that was unorganized, chaotic, etc of which there is an infinite amount.

Along the same lines, the Doctrine and Covenants, which is regarded as canon in the LDS Church, teaches that "the elements are eternal" (D&C 93:33), and that matter "was not created or made, neither indeed can be" (D&C 93:29) (in accordance with the first law of thermodynamics).

As for how this applies to God, the Doctrine and Covenants teach that "The Father has a body of flesh and bones as tangible as man's." (D&C 130:22). Bodies are made of matter, and matter is energy, thus God is made of energy, just like you and I are.

There are too many different ideas about the relationship between God and the Universe to cover them all. Here's an attempt to explain one fairly common view (which seems to have been offered in other answers):

If you have any background in programming: Suppose the entire universe that we are aware of is actually a simulation running as a program on a computer. Now suppose the programmer opens up a debugger and starts fiddling with variables. He or she might mess up invariants or relations that the program would originally obey, but you wouldn't call the programmer and the debugger part of the program. The debugger wouldn't have simulated energy, nor would the programmer (unless they specifically programmed it to work that way)--they're outside of the program (presumably with real energy).

Analogously, one can envision God as the programmer who can intervene as desired or not; He isn't really part of the simulation, so to speak, so even though everything that is being simulated has simulated energy, He does not intrinsically, and His interventions may or may not depending on the details of what happens.

Energy is merely matter in another form (see Wikipedia). Matter and energy are both part of the created universe, made by God (see Book of Genesis). Energy, like matter, was created by God. It is not an attribute of God, any more than height, weight or hair colour.

Obvious exception for the incarnated Jesus, who would have had energy, height and hair colour.