Most demons(from my understanding) eat or survive off a negative human emotion. Some examples would be Evelynn who’s the demon of agony, nocturnal is the demon of mortal terror and swains arm is the demon of secrets. So my question is what is tahm kench the demon of?
Most likely Gluttony. His theme of eating and taste and stuff make sense for that.
I think Riot is trying to do a seven deadly sins type thing with their demons, eventually, with Tahm being Gluttony and Eve being Lust.
Deals, probably. That or greed/gluttony/pride. He's essentially a crossroads demon. People make deals with him for their own personal gain, and then he collects.
"The difference, as I understand it, is this:
Gluttony (gula) is impulsive overindulgence in something which is otherwise normal or necessarily. Like lust, it's fundamentally a failure of self-control, but unlike lust, it concerns something which is not inherently bad or out of bounds (everybody has to eat, after all), but it becomes wrong through the manner, frequency, or degree of consumption. The term is mostly used in reference to food and drink (including alcoholic beverages), but you can apply it to just about anything (e.g. binge-watching television or donating to charities).
Greed (avaritia), also known as avarice, is simply insufficient generosity. It's the application of inconsiderateness or lack of compassion to a specific situation: keeping what you have when it would be better to give it away. In other words, it's a sort of stinginess or miserliness. Whereas gluttony is a failure of self-control, greed is a failure of foresight or consideration, or is a result of a compulsion to collect or acquire things, or a misplaced sense of competitiveness with others (although that shades into envy).
So you might say that gluttony involves having too little sense of self, while greed involves having too much. A glutton fails to ration or conserve their resources (including health and dignity), while a greedy person succeeds all too well at conserving them, hoarding them even, at a cost to others."
I think this is a good answer.
ambition or desire would probably fit.
TK feeds on mortal misery. Glutonny, greed, desires etc arent his food, those are simply ways to get food for him, in the same way that Evellyn doesnt feed on lust but use it as a tool to cause pain. Those quotes will probally clarify it:
Known by many names throughout history, the demon Tahm Kench travels the waterways of Runeterra,** feeding his insatiable appetite with the misery of others.**
So basically, Tahm Kench uses greed and desires as a tool, but he ultimately feeds of misery.
All the Dev/writer talk I'd read so far implied to me he was a despair-demon. He uses all his powers and methods to invoke despair in his prey, and feeds on /that/, and he is very gluttonous for his feeding on said despair.
His launch-story was him literally setting up a guy's entire life for success, just to take it all away, including his love, so that the despair would be at maximum pitch, and that was a true 'feast' for him.
The physical eating is just a tool he uses to accomplish that.
I think they said he feeds off misery. That's why he gives people exactly what they want and then takes it all away when they're most happy
It is gluttony if your going by fact-based stuff since in-game he mentions this himself and uses this term, but overall greed could be the term applied, not just gluttony, but greed, and pre-determined deals and attacks that satisfy him.
I choose to believe he is the demon of greed, and I believe he does the best job of being a "demon" than the others, they just don't seem to fit the bill all that well, they aren't purely what they wish to be, it is just part of their character, similar to Swain, Darius, Noxus, and all of their connections to strength, or Urgot with Pain, I feel like all the other "demons" are just as closely tied to the thing they are supposedly demon of, as Urgot and others are tied to their themes. Tahm Kench, however, is purely and inherently a perfect embodiment of greed, it isn't just part of his goals, mindset, character traits, it is shown in his appearance, his actions, his style, and etc. but he doesn't push the theme too hard like the rest, he just is, greed.
This is why I once mained Kench, and although I couldn't stick with the same champion forever (it gets old ya know? even though he is great...) I still go back every now and then and make it my mission to Unbench, The Kench. 
He's a demon of greed but that doesn't mean he isn't gluttonous, and honestly greed and gluttony are pretty similar anyway. Greed is a desire for more, so gluttony is kind of a subdivision of greed (imo).
He traps people in deals that they freely could have refused but their desire for something more, another night partying, another chance to gamble, another experiment to sell, leads to their downfall.
Pride kind of works too, but not as much, though he does tend to build up his victims before knocking them down, so maybe.
