Can you separate CO2 into carbon and oxygen using pyrolysis?
Standard enthalpy of formation of carbon dioxide is -393.5 kJ/mol. Standard enthalpy of formation of both oxygen and graphite is 0 J/mol. Standard entropy of carbon dioxide is 213.8 J/mol-K (NIST). Standard entropy of oxygen is 205 J/mol-K (Chem LibreTexts). Standard entropy of graphite is 5.69 J/mol-K (ChemLibreTexts).
Everything below is a lot of fairly inexact approximations/assumptions because I didn't want to use a better model or search literature for empirical adjustments and/or constants.
Assuming constant T,P (which I doubt would be true under reactor conditions) G = H - TS. Approximating $\Delta H, \Delta S$ as constant with T, $\Delta G = (0 - (-393500)) - (T \cdot (5.69 + 205 - 213.8)) = 393500 + T \cdot (3.11)$. Then $\Delta G &lt; 0$ regardless of temperature. Catalysts also do not affect thermodynamics, so this process can never occur spontaneously. Increasing pressure of the reaction chamber would not help appreciably either, as the reaction has 1 mol gas on each side. While assuming $\Delta H, \Delta S$ are constant with T is quite incorrect as conditions become more extreme, the approximation should suffice to show that the reaction does not occur easily. Additionally, even if one were to account for the perturbations in $\Delta H, \Delta S$ with T and P, these conditions would be expensive to maintain in a reactor. (Note that this analysis does not even account for the activation energy/kinetics problem! I did not look up the activation energy of the problem, but it is entirely possible that the extreme conditions would still not allow the process to occur at a reasonable rate.)
As a result, this reaction is (probably) practically only driven forward by direct energy input (aka a coupled reaction/external energy). This often, if not always, involves some energy input, such as light or a coupled chemical reaction, that directly impacts the electron states of the participant compounds. These changes in electron states change the identity/properties and stability of compounds, and, when applied effectively, they can help drive the reaction forward. However, relevant changes to electron states using heat not only often involve very high temperatures, but the effect of heat on electron states is also a highly statistical process. Thus, it may be relatively inefficient. Thus, pyrolysis, while it may be theoretically possible, would be incredibly impractical at achieving separation of carbon dioxide. (Also, to compare to your methane example, methane combusts readily, carbon dioxide does not. Methane has a standard enthalpy of formation of -74.87 kJ/mol.)
Finally, if you still need more reasons to look at a different way to drive this reaction, I have not mentioned: energy/cost to continuously scrape/collect the produced carbon, purify/collect/pressurize an inlet carbon dioxide stream, (if you actually want to run the reaction, you'll probably want) catalytic surfaces to encourage carbon deposition, possible packed/fluidized bed catalysts, heat exchangers, and much, much more.
ELI5: Methane combusts/reacts readily and is an &quot;unstable&quot; compound. Carbon dioxide is very stable and does not react easily. Higher temperatures can make some reactions that do not happen at a lower temperatures occur, but this particular reaction receives no such benefit. Heat and catalysis can also drive forward the speed of a reaction, but the reaction has to be able to occur first. Pyrolysis, as you described it, involves catalysts and heat. As a result, pyrolysis cannot be used to separate carbon dioxide into carbon and oxygen.
tldr: Excepting conditions extreme enough to perturb $\Delta H_{rxn}, \Delta S_{rxn}$, conditions of constant T,P never favor breakdown of carbon dioxide into elemental carbon and diatomic oxygen. Pyrolysis cannot drive this reaction.