Here's an analogy, my own, that I checked out years ago with a very good GR expert. His comment was roughly "that may be one way to think of it," so please take it with some caution. I also immediately bow to anyone with deep tensor knowledge. Still, it is an analogy that is a lot easier to get hold of than most, so I think it's worth describing in answer to your question.
GR says that gravity has a deep equivalence to acceleration. Thus if you imagine a large, flat plane accelerating through space perpendicular to the surface, that flat area has an close mathematical similarity to a small, almost-flat section of a very large sphere with gravity acceleration comparable to that of the sheet.
Notice that for the flat sheet, the acceleration vectors for any point on the sheet are exactly parallel. That's important, because as with velocity vectors, acceleration vectors that are exactly parallel essentially hide the energy that they contain from each other. Thus if two cars are moving down the highway very close together at the same speeds and wit precisely parallel paths, someone can step from one to the other without any dangerous release of energy. The same is true if both cars are accelerating with identical vectors, although of course in that case you would need a ledge to walk on because of the gravity-like field.
Both cases are in sharp contrast to the opposite extreme of two cars with velocity or acceleration vectors that result in head-on collisions. In those cases, the "hidden" energy becomes very real indeed, and catastrophic for anyone inside the cars. In between those two extrema you have anywhere from a "tiny" bit of collision energy from slightly non-parallel car vectors to increasingly drastic energy releases as the parallelism fails more completely.
Now, let's go back to the accelerating sheet in space. There your acceleration vectors are perfectly parallel, so as with the example of the cars, the energy implied by those vectors is hidden from people on the sheet.
I also mentioned that the sheet is almost like a section of the surface of a world with gravity that gives the same acceleration, but of course there's an important difference: The world is a ball, so the gravity vectors on its similar sheet are not exactly parallel; they diverge by a tiny bit.
Now that has an odd implication, which is this: When mass-generated gravity fields curve around on themselves -- and of course all of them do -- they also create a bit of a mismatch in acceleration paths that evidences itself as a bit of energy added to any object that gets caught between its very slightly divergent acceleration paths. This "jostling" energy will be proportional to the area of the object that is orthogonal to the (average) acceleration path. It is of course astronomically tiny for gravity fields such as earth's, yet at the same time it is there and is inherent in the curved structure of the field.
Now finally, imagine that the radius of the object with the gravity field shrinks while the gravity field itself stays constant. In that case, the earlier approximation of the accelerating sheet gets mapped into an increasingly and in the end overtly curved subsection of the spherical surface of the object. At the same time, the acceleration vectors become so non-parallel that even for a small bit of matter, the acceleration vector at one edge of the matter becomes noticeably non-parallel to the other edge. The vectors now have a significant "jostling" factor that is inherent in space itself, and which will express itself on any matter within that space at a higher temperature.
As an addendum, I would add that tidal forces can be interpreted as the vertical version of the same effect. That's because the acceleration vectors will also change in magnitude (but not direction) from the top to the bottom of the object. The simplest heuristic for realizing that more analysis is needed is to look at the size of the object relative to the geometry of the gravity field. If field is strong and the object is large enough to "see" changes in the geometry or strength of the gravity field acceleration vectors, there's going to be trouble, and you need to do a precise mathematical analysis.
So, I think horizontal acceleration vector divergences give a reasonably accurate portrayal of why there is a temperature component to curved space. I'm also pretty sure that there is an equivalence to the concept of a temperature at the surface of a black hole, which as in the case I just described is also extremely dependent on how sharp the curvature is at the surface of the black hole.
And again, while I'd probably say that I'm better than average at four dimensional geometry problems, I am no GR expert and make no claim to be one. My only verification of my analogy was that quick look from an extremely good GR expert years ago who was kind enough to take a look at it.