Civilization would develop pretty much the same as it did.
First I upvote AlexP's comment: The uniform climate is across the YEAR, it is not uniform from the north pole to the south pole. So there is still some variation, and that civilization started in warm climates anyway; the middle East basically.
If the entire Earth is habitable and arable, the same organization would occur. Hunter gatherers most likely accidentally invented farming, by carrying fruits and edible plants collected in one place with them, and discarding the rinds or seeds on fertile ground. Since many HG groups make circuits of a large area of land; all it takes is a normal human memory to realize that primitive versions of squash or cucumbers or watermelon or cantaloupes are growing exactly where you ate the same thing a year ago. And spit out the seeds. Eureka! The invention of intentional farming.
Which could be accomplished still by HG, or herders (an evolution of HG into tribes with goats, sheep or cattle they drive on a circuit to consume fresh vegetation). They can scatter seeds at each campsite in their circuit; and move on to the next: Hopefully harvesting some of whatever they scattered there last year, on their previous visit.
But eventually farming becomes "institutional" in the sense that a tribe creates a permanent settlement and plants a large amount of high calorie food (like grain) and stays to protect it. Probably a herding tribe; since they can also protect land for grazing, build corrals and more permanent shelters (instead of tents), making animal tending easier and life more comfortable. Also, 95% of food plants like grains are not consumed by people, but will be consumed by goats. (Goats can eat and get energy from nearly all vegetation, including roots, stalks, rinds and even branches, and goats are not picky at all. Plus they produce high calorie milk and meat: Making them an ideal first domestic animal.)
Specialization (of jobs) and Civilization springs forth. Probably even faster and in more places than it did on Earth, where we had the complicating factor of dealing with seasons, winters, and harsh summers.
The seasons did not drive us to form settlements. We formed settlements because we found much easier ways of getting meat (herding) and much easier ways of getting calories (farming) and settlement was much easier than carrying everything five miles every day. Once you figure out how to get your calories with very little effort, you stop working so hard to get calories, and start working on other things. Like music, then flush toilets, then inventing the Internet. (In that order.)