1) Problem with the energy of causes
Energy-neutral causation need not involve anything supernatural or extraordinary. A rock on the tracks may cause a train to derail, but it does not expend any energy to do so, it simply redirects it. So does a rail switch. Reflecting on these examples shows that there is a problem with associating energy with causes in the first place, it mixes two incompatible notions of causation. The Cartesian "folk" notion is modeled on a somebody/thing (the "agent") standing over and above the causal chains and altering them by exerting "energy" to push things along. Cognitive psychologists attest that this is how infants grasp the "metaphysical" concept of causation, see e.g. On the Birth and Growth of Concepts by Mandler:
"Michotte (1963) showed that the timing of the launching events in which one billiard ball hits another is crucial for our illusion that we see one ball ‘make’ the other move. We see a causal relation when a conflict exists between two types of continuity cues... Of course, there is more to an adult concept of cause than one object launching another. However, as Leslie showed, infants are attuned to contact versus no contact between moving objects."
This is associated this with the physical exertion and direct contact required to get the ball rolling, and the seed of the folk concept of (efficient) cause is planted.
2) Physical causation and energy
In contrast, the physical concept of causation is essentially Humean, there is no such thing in the ontology. It mimicks the folk notion by using counterfactual alterations of "initial" conditions: event X is a (partial) cause of Y, if when we alter the initial conditions to exclude X the probability of Y goes down. This is the gist of the prevailing Lewis's theory of causation, and it clearly relates to our probabilities, not to something out there. "What caused Y?" is a purely pragmatic question of singling out some X from the background (presumed fixed), but for which Y would not have happened. The distinction between "the cause" and "the background" has no ontological significance, they are simply of different importance to us.
When the background is taken into account the "causes", at best, can be said to transfer energy to the "effects", they do not stand over the stream of events to inject it. This is what energy conservation amounts to. Feynman famously exploited the idea by proposing reversible quantum computers that perform computations without expending much energy. Incidentally, this is in agreement with the Aristotelian scheme, where the only source of energy is the prime mover, everything else just passes it along (unless we take some vague passages in De Anima to mean that the "divine in all of us" agent intellect somehow partakes in the prime mover).
But we can explain what the intuition is getting at. Event X amounts to some object(s) forming a particular configuration favorable to Y occurring down the road. At first, it seems that to be causally efficacious it must pack some energy that contributes to effecting Y. But the example of a rail switch shows that this need not be so - it may instead act as a trigger, that redirects motion, or a catalyst, that facilitates a chemical reaction without getting spent. One may object (as PeterJ did in the comments) that it still takes energy to place the trigger/catalyst into a strategic position. That is true, but it is quite different from the energy that it contributes as a "cause" to its "effect". The placement energy is, in fact, the energy of agent cause, which must be contemplated if we take the counterfactual alterations as really effected. However, it turns out, as we shall see next, that even that energy can be zero in some circumstances.
3) Agent causation without energy
To speak of the energy of causes one needs to reify the counterfactuals, as the folk notion does. Somebody/thing must alter the initial conditions in re, i.e. we must admit agent causation. The "agent" need not be anthropomorphic, it can even be inanimate, but must be able to alter causal chains from the "outside" (naturally, this is linked to objective chance and free will). Can it do so without expending energy? The surprising answer is yes, in both classical and modern physics. Classical systems with non-Lipschitz forces admit initial states where the solution to the equations of motion is non-unique, the "agent" can then "pick" one, while complying with all the conservation laws. The well known example is the ball sitting at the top of the Norton Dome, which is free to "choose" the direction to roll down. The example was already known to Boussinesq in the 19th century, whom Maxwell praised for linking the idea to free will, see History of the study of indeterminism in classical mechanics:
"It may at any instant, at its own sweet will, without exerting any force or spending any energy, go off along that one of the particular paths which happens to coincide with the actual condition of the system at that instant. In most of the former methods... there was a certain small but finite amount of... trigger-work for the Will to do. Boussinesq has managed to reduce this to mathematical zero..."
This is somewhat exotic, but quantum mechanics provides a more straightforward means of accomplishing energy-neutral causation, by altering the probabilities of the wave-function collapse. This is the basis of Eccles's model of the agent causation by the human brain:
"Without violation of the conservation laws quantum selection is the only possible way of producing different final states from identical initial conditions in identical dynamical situations, and thus with the same values of the conserved quantities... We put forward the hypothesis that mental intention becomes neurally effective by momentarily increasing the probabilities for exocytoses in a whole dendron and, in this way, couples the large number of probability amplitudes to produce coherent action."
Esfeld in Is Quantum Indeterminism Relevant to Free Will? gives a critical discussion of Eccles's (and other "quantum mind" Cartesian interactionism) proposals. The problem is that while this mechanism escapes violating the conservation laws it still requires violating physical laws, namely the probability distributions prescribed by quantum mechanics. And, at present at least, we have no empirical evidence that they are so violated:
"If intentions, as conceived in the framework of interactionism, are to exert a regular influence on physical events, the probability rules of the physical theory in question do not indicate the complete probabilities of these physical events... In short, instead of having to endorse an additional force for metaphysical reasons, we have to endorse a change to the probabilities that a physical theory indicates for metaphysical reasons."
Be it as it may, even agent causes can act effectively without any energy costs. What is required of such causes is injecting information, objectively conceived, into the stream of events, not energy.