My current author bio mentions an altar to Cthulhu; so, it’s perhaps no surprise that someone asked me whether I’d worship the gods of the Mythos if they existed. The answer is: it depends on what you mean by worship.
The first question of course is whether there are gods in the Mythos at all. Some critics point out that Lovecraft describes beings that—from the perspective of humans—have incomprehensible abilities and desires but are still subject to the vagaries of an uncaring universe, so are aliens rather than gods. However—although the existence of limitations is true—omnipotence isn’t an essential quality of things we call deities (for example, many of the Norse gods lose items of power, are defeated in various ways, or die in Ragnarök); so we can say that the beings of the Mythos aren’t gods like the Abrahamic god but having weaknesses doesn’t make them not gods. And, assuming they are technically aliens, there’s also the issue of whether it matters at the human scale: if a being can grant the ability to defy physics as we understand it to those who follow instructions and displays power we can neither comprehend nor effectively counter save with certain rituals, then we interact with it in the same way religion interacts with gods; so is there a practical use to dividing between god-proper and god-like? For this post, I’ll assume there isn’t and “god” includes anything worshipped by humans even if it might arguably be an alien.
The second is what worship means. While worship is often used in English to mean any pattern of religious ritual or service, it carries an implied sense of “reverent love or devotion”, a sense that might stretch as far as being “unconditional”. One can obey for reasons other than loving respect. There are, for example, cultures that have dark gods who are propitiated as a matter of good sense but not loved. Thus, one isn’t limited to either worshipping or rejecting gods.
So, would I worship Mythos entities? Almost certainly not: even leaving aside those whose interactions don’t seem comprehensible, many of the things they do or seem to demand of their followers inspire horror rather than love.
Would obey them? Potentially. History is full of people choosing to perform acts due to a threat of force and I’m self-aware enough to accept that there are almost certainly things that mildly offend my freedoms that I wouldn’t die to avoid doing.
If, however, one takes the wider meaning of worship as repeated action not supported by current science, then it’s a trickier question. Yog-Sothoth in its form as ’Umr A-Tawil seems benevolent so, while I don’t feel a reverent love for it, I might perform metaphysical rituals I didn’t understand (i.e. magic/prayer/hyper-science) in exchange for benefits. Nyarlathotep, while mostly considered evil, might be demonstrating a path to transcend humanity that—in hindsight—isn’t the evil it seemed, so I could conceive of being grateful to it after the fact.
Of course, if we’re to believe the Gospel of Leman Cthulhu wants us to fear him not love him anyway.