*EDIT*
Damn, this is a really long post! I apologise for its lack of brevity! Hopefully this will explain how it could theoretically be possible to have something like nethercaps actually do the things they do.
*/EDIT*
Boiled down:
Thermodynamic laws.
1) in a closed system, the change in internal energy is equal to the amount of heat, minus the work donne.
2) in a closed system, entropy can only increase or stay the same, but never decrease.
3) the entropy of a perfect crystal at absolute zero is exactly 0.
Assertions:
The cavern system is not closed. Energy is continually added to the system via a semi-steady influx of heat, light, and chemical energies from volcanic processes. Excess thermal waste bleeds out of the system through passive diffusion through the surrounding rock. Nearby volcanism injects hydrogen sulfide (SH2),sulfur dioxide (SO2), and carbon dioxide (CO2) gasses, which permeate heat fractured rocks along depositional layers and faults. A slow but steady stream of these gasses enters the ecosystem from the floor of cavern layer 3. In addition, there is a surplus heat gradient of cavern layer 3, vs cavern layer 2. Heat flows from layer 3 to layer 2, in the attempt at reaching equilibrium. This disequlibrium can drive thermal processes to perform work. (See carnot heat engines, such as sterling engines. See also the "overunity" LED pointed to earlier.)
Observations:
Nether caps lower surrounding temperatures, creating a thermal gradient. The law of energy states that energy is never destroyed. This means the thermal energy is converted to some other form for the thermal gradient to exist. Since the nether cap does not emit energy of any known form, the energy must be stored in some fashion, such as in chemical bonds. This implies an endothermic metabolic process. In order to create a thermal gradient, energy must be expended to perform this sequestration. This energy cannot logically come from heat, because it is being used to move heat, unless a gradient of some sort is already existing that can be exploited. (In this case, however, the induced gradient cannot be greater than the natural one powering it.)
Nether caps grow in cavern layer 3, which has the highest concentration of chemical energy from the volcanic vent. The chemical energy from the volcanic vent supplies the energy needed to create a strong thermal gradient with which to do useful work. This implies the consumption of several chemical reactants for energy, to create a reaction gradient favorable to the synthesis of the high energy compound. This reaction must be exothermic.
This means:
1) nether caps respire osmotically, absorb SO2, CO2, and H2O, and synthesize H2SO4 as a primary source of chemical energy. Reaction of H2SO4 with metallic ions in the bedrock can provide a low level ion pump, which can drive other reactions.
2) nether caps use this energy from the chemotrophic metabolism of sulfur and water along with the low grade electrical potential of the sulfuric acid waste product interacting with the mineral substrate to fascilitate the synthesis of more complex molecules for long term energy storage. This synthesis reaction makes use of the absorbed CO2, is endothermic, and uses a methodology similar to the prevoisly cited "over unity" LED. More chemical energy is stored than is produced by the chemical respiration of the sulfur cycle, by passive absorption of thermal energy.
3) the stored chemical energy of the long term energy-rich storage compound is released to further promote heat energy "consumption" via a secondary exothermic respiration reaction.
4) the energy released by exothermic reactions is less than the thermal energy absorbed, by exploitation of the energy state thresholds of the enzymic catalysts used. (In the case of the LED, a low level electrical current just barely enough to bridge the bandgap of the semiconductor produces more photons than the source input should normally allow. The difference in photon output comes from the random fluctuations in the baseline of the junction caused by thermal noise. The thermal noise is just enough to push the valence electrons across the band gap, releasing the energy as a photon. The temperature of the junction drops proportionally to the energy of the photon emitted. The nethercap, however, uses a semiconducting enzyme near the cellular membrane that makes use of the ionic potential between the internal H2SO4 concentration, and the metallic ions outside, to prime itself as the current source. Passive thermal absorption by the enzyme is then converted into useful work by adding additional electrons to its substrate. Rather than emit the energy as photons, it emits the energy by pushing a higher energy valence electron out of place, catalyzing a higher energy reaction than it would otherwise be able to do. This is similar to what chlorophyll does when it absorbs photons to push electrons into higher energy states to form sugars.) Metabolism of the stored energy molecule is used to increase enzyme density, concentrate absorbed fuel gasses, and improve internal cellular efficiency. This increases the rate of thermal conversion, favoring sequestration of energy rich compounds.
This places bounds on the degree of induced thermal gradient that nethercaps can create, however. Outside of the "sweet spot" used by its enzyme, nethercap metabolism would be adversely effected. This means nethercaps would not be able to handle extreme temperatures. (They would want the environs to be a very precise temperature range. Outside of that range, they can't do their thermodynamic trickery.) This may further explain nethercap specificity to cavern layer 3. (Higher layers are too cold!) If nethercaps absorbed energy too quickly, they would suffer from being too cold. It is likely that the "coolness" of nethercaps is likely only a few degrees cooler than the bedrock they grow on.
Rate of energy sequestration would be directly tied to the rate of sulfur respiration, and the degree of electrical potential that could be induced between the cytoplasm and the mineral bedrock, with bounds set by the thermal reactivity range of the sequestering enzyme, and surface area used for the reaction.
In contrast to green plants, where complex synthesis happens in the leaves, the nethercap electro-thermalsynthetic reaction would take place in the "roots", where the electrical charge gradient is the highest. Being a mushroom, this means the reaction occurs in the mycelium.
The fruiting bodies of mushrooms form as the result of two or more colonies of genetically heterogenous mycelium coming into contact with each other. Mycelial fibers from both colonies act on each other through hormonal signals to rapidly grow, and produce a fruiting body. Within the fruiting body, gametes from each culture combine to create the spores that develop inside. Due to this, energy is concentrated in the fruiting body.
For this reason, I would personally expect nethercaps to make the ground cooler, but to be slightly exothermic in their fruiting bodies, like other fungi. The difference would be that nethercaps would be capable of actually adding energy to an ecosystem instead of consuming it.
That nethercap wood would be inducing a strong gradient at all defies rational explanation, given what little I know about chemistry and thermodynamic systems.