p-Cymene , according to PubChem, may be metabolized to metabolites were formed: monohydric alcohols, diols, mono- and di-carboxylic acids and hydroxyacids. Conjugation with glycine of the cumic acid formed was extensive in guinea pigs., Metabolites in other studies include p-cresol and cumic acid. When looking for p-cymene targets we must also consider potential metabolites as well as oxidation products.
Cumin essential oil was found to contain cuminaldehyde (44.53 %), p-cymene (12.14 %), (-)-β-pinene (10.47 %) and γ-terpinene (8.40 %).  According to a Slovakian and Polish group athe main components of T. vulgaris essential oil are thymol (48.1%), p-cymene (11.7%), 1,8-cineole (6.7%), γ-terpinene (6.1%), and carvacrol (5.5%).  A Japanese group found that p-symene was 47.2% of the essential oil from Dysphania ambrosioides,also known as Jesuit’s tea.  The pinene like terpene askaridole composed 35.7% of this essential oil.  Note that cuminaldehyde and thymol are oxidation products of p-cymene.
In an Italian study, thymol was found in previous studies of the authors to improve gastrointestinal health in piglets; therefore, a followup study showing effects of thymol on the G.I. endocannabinoid system was conducted..  Probably the actual thymol receptor is TRPA1.  The Tocshi survey of endocannabinoids will take us on an interesting twist that suggests yet another target: the mitochondria for p-cymene are related terpenes. 
One hundred sixty male and female weaned piglets (Duroc × Large White) were assigned to five groups of 32 each.
- (1) control group fed the basal diet
- (T2) basal diet + 25.5 mg of thymol/kg feed
- (T3) basal diet + 51 mg of thymol/kg feed
- (T4) basal diet + 153 mg of thymol/kg feed
- (T5) 510 mg of thymol/kg feed….10x highest recommended dose 
Note that all the target mRNAs different than the T1 control group were greater than that group. 
Ion channels ignored
This post is ignoring many fine studies suggesting that p-cymene gates potassium and calcium ion channels. These ion channels are being ignored because of the unrealistically high concentrations needed to see an effect.
The mitochondria and energy balance
Another study examined the affect of p-cymene on isolated rat mitochondria.  The premise was that bacteria are like mammalian mitochondria. If a compound is antimicrobial, might it also be cytotoxic to mammalian mitochondria? Isolated rat liver mitochondria were used in this study.  The paper requires that we abandon the notion that the ATP synthase is part of the electron transport chain. Electron transport stops at complex IV. This Rice University page gives a nice overview. The Rice U authors introduce the concept of the ADP:O ratio, entropy, and heat generation. The mitochondria are not only generators of ATP but also heat. The Rice authors didn’t really elaborate as to whether conformational changes in complex I-IV and the F1F0 ATPase upon binding phosphate and ADP could be communicated via changes in the lipid bilayer. This is when our story intersects the cannabinoid pathway(s). The endocannabinoids tend to turn off heat generation by the uncoupler protein.
- State 2 Respiration, the electron transport chain up to complex IV. p-Cymene at 25-100 nmol/mg protein had no effect.
- State 3 Respiration, Though not part of the electron transport chain, the F0F1 ATPase, aka complex V, can speed up respiration in the presence of ADP. A slight decrease was seen at 75 nmol p-cymene per mg protein.
- State 4 Respiration, oxygen consumption in the absence of ADP. A slight increase was seen at 50 nmol per mg protein.
Note that 75 nmol per 1mg protein is not 75 nmol/Liter. In any mammalian cell, the p-cymene has to diffuse through the cell membrane first.
The respiratory control ratio, State 3/ Sate 4, was found to decrease by 50% in the presence of 100 nmol p-cymene per mg protein. A significant decrease was seen at only 25 nmol/mg protein. 
Oligomycin is an antibiotic that prevents H+ transport by the ATP synthase pore. Oligomycin has no effect on state 4 respiration but inhibits state 3 completely… without direct effects on the H+ gradient. In agreement with the Rice University page on isolated rat liver mitochondria, Custódio found that in the absence of p-cymene, oligomycin had no effect on state 4 respiration. In the presence of 50 nmol p-cymene per mg protein, oligomycin inhibited state 4 oxygen consumption. 
The higher concentrations of p-cymene, 75 and 100 nmol/mg, increased state 2 respiration in the presence of ATP.  ADP must bind to the ATP synthase for H+ to flow through the pore.
The lag phase is the time it takes to produce ATP after ADP has been added.  p-Cymene does not increase osmotic swelling (or shrinking) due to an excess of sucrose on the outside. Ca2+ permeability is also not affected. Cyclosporin
Ca2+ was used to open the mitochondrial transition pore. The cyclic peptide cyclosporin was used to close it. Part of the image below was adapted from Semantic Scholar. VDAC is the voltage dependent anion channel. Parts of p-cymene and related terpenes resemble the leucine side chains of cyclophorine. Any interaction p-cymene with cyclophilin D is negligible.
The interaction of thymol with membranes has been covered on this site. p-Cymene, thymol without the hydroxyl group, maybe predicted to penetrate deeper within the lipid bilayer. Custódio and coworkers name thymol, carvacrol, and γ-terpinene as anti-bacterial terpenes that may also poison the mammalian mitochondria.  It is interesting to note that p-cymene tends to increase mitochondrial uncoupling. Endocannabinoids tend to turn off expression of the uncoupler protein. Thymol increases expression of proteins that would promote mitigation of the action of p-cymene, and possibly related terpenes. Perhaps the endocannabinoid system up regulation by thymol, and possibly related terpenes, is an evolutionary response to cross-over anti-bacterial effects on mammalian mitochondria.
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- Dougnon G, Ito M. Role of Ascaridole and p-Cymene in the Sleep-Promoting Effects of Dysphania ambrosioides Essential Oil via the GABAergic System in a ddY Mouse Inhalation Model. J Nat Prod. 2021 Jan 22;84(1):91-100.
- Toschi, A., Tugnoli, B., Rossi, B., Piva, A., & Grilli, E. (2020). Thymol modulates the endocannabinoid system and gut chemosensing of weaning pigs. BMC veterinary research, 16(1), 289. PMC free article
- Kaji I, Karaki S, Kuwahara A. Effects of luminal thymol on epithelial transport in human and rat colon. Am J Physiol Gastrointest Liver Physiol. 2011 Jun;300(6):G1132-43 free article
- Custódio, J. B., Ribeiro, M. V., Silva, F. S., Machado, M., & Sousa, M. C. (2011). The essential oils component p-cymene induces proton leak through Fo-ATP synthase and uncoupling of mitochondrial respiration. Journal of experimental pharmacology, 3, 69–76. PMC free article