Consider a system composed of chemical species (e.g. water splitting) in thermodynamic equilibrium at constant pressure and thermodynamic temperature T:
:::::HO(l)  H(g) + 1/2 O(g)    (1)
Equilibrium is displaced to the right only if energy (enthalpy change ΔH for water-splitting) is provided to the system under strict conditions imposed by thermodynamics:
* one fraction must be provided as work, namely the Gibbs free energy change ΔG of the reaction: it consists of "noble" energy, i.e. under an organized state where matter can be controlled,  such as electricity in the case of the electrolysis of water. Indeed, the generated electron flow can reduce protons (H) at the cathode and oxidize anions (O) at the anode (the ions exist because of the chemical polarity of water), yielding the desired species.
* the other one must be supplied as heat, i.e. by increasing the thermal agitation of the species, and is equal by definition of the entropy to the absolute temperature T times the entropy change ΔS of the reaction.
Hence, for an ambient temperature T° of 298K (kelvin) and a pressure of 1 atm (atmosphere (unit)) (ΔG° and ΔS° are respectively equal to 237 kJ/mol and 163 J/mol/K, relative to the initial amount of water), more than 80% of the required energy ΔH must be provided as work in order for water-splitting to proceed.
If phase transitions are neglected for simplicity's sake (e.g. water electrolysis under pressure to keep water in its liquid state), one can assume that ΔH et ΔS do not vary significantly for a given temperature change. These parameters are thus taken equal to their standard values ΔH° et ΔS° at temperature T°. Consequently, the work required at temperature T is,
As ΔS° is positive, a temperature increase leads to a reduction of the required work. This is the basis of high-temperature electrolysis. This can also be intuitively explained graphically.
Chemical species can have various excitation levels depending on the absolute temperature T, which is a measure of the thermal agitation. The latter causes shocks between atoms or molecules inside the closed system such that energy spreading among the excitation levels increases with time, and stop (equilibrium) only when most of the species have similar excitation levels (a molecule in a highly excited level will quickly return to a lower energy state by collisions) (Entropy (statistical thermodynamics)).
Relative to the absolute temperature scale, the excitation levels of the species are gathered based on standard enthalpy change of formation considerations; i.e. their stabilities. As this value is null for water but strictly positive for oxygen and hydrogen, most of the excitation levels of these last species are above the ones of water. Then, the density of the excitation levels for a given temperature range is monotonically increasing with the species entropy. A positive entropy change for water-splitting means far more excitation levels in the products. Consequently,
* A low temperature (T°), thermal agitation allow mostly the water molecules to be excited as hydrogen and oxygen levels required higher thermal agitation to be significantly populated (on the arbitrary diagram, 3 levels can be populated for water vs 1 for the oxygen/hydrogen subsystem),
* At high temperature (T), thermal agitation is sufficient for the oxygen/hydrogen subsystem excitation levels to be excited (on the arbitrary diagram, 4 levels can be populated for water vs 8 for the oxygen/hydrogen subsystem). According to the previous statements, the system will thus evolve toward the composition where most of its excitation levels are similar, i.e. a majority of oxygen and hydrogen species.
One can imagine that if T were high enough in Eq.(3), ΔG could be nullified, meaning that water-splitting would occur even without work (thermolysis of water). Though possible, this would require tremendously high temperatures: considering the same system naturally with steam instead of liquid water (ΔH° = 242 kJ/mol; ΔS° = 44 J/mol/K) would hence give required temperatures above 3000K, that make reactor design and operation extremely challenging.
Hence, a single reaction only offers one freedom degree (T) to produce hydrogen and oxygen only from heat (though using Le Chatelier's principle would also allow to slightly decrease the thermolysis temperature, work must be provided in this case for extracting the gas products from the system)