Gravity causes objects of a higher altitude to fall to a lower altitude.
Thermodynamics, similarly, causes heat to flow from areas of higher temperature to areas of lower temperature.
Now, it’s possible to have an object suddenly “pop up” in altitude like a ski-jumper or water in a fountain. Is it also possible for an analogous local reversal in thermodynamics?
Sure.
It’s called an “Air Conditioner”
Maybe I’m misinterpreting what you mean, but I think you’re talking about quantum fluctuations. Heisenberg’s Uncertainty Principle allows for energy to exist for short periods of time. These cause virtual particles to appear in a constant froth underlying all reality. By statistical distribution some of these energy fluctuations would be so large that they would appear on the macro level, although the entire existence of the universe so far probably isn’t long enough to have a noticeable one appear.
That’s the only thing that can suddenly “pop up” that I’m familiar with, and it doesn’t involve gravity. What might is that an object already in existence could suddenly move up in altitude as an analogy, I suppose.
And to get to thermodynamics, there is also the analogy of a cold pot of water suddenly boiling at the surface if all the hot (i.e. most rapidly moving) molecules happened to group together and be collectively above the boiling point. That would also require more than the universe’s time to happen, but it statistically possible as an extreme outlier of distribution.
You’re confusing two different flavors. It is possible for an object to convert kinetic energy into gravitational potential energy (ski jumper). Temperature is not a form of energy. However, an object’s internal energy is related to its temperature, and so there is a semi-analogy here. Let’s say I convert electrical energy into heat exchange, bringing two objects out of thermal equilibrium. The object that gets hotter now has more thermal energy and more temperature, and in the absence of artificial heat exchange, the two objects will then reach equilibrium again in short order.
In other words, an air conditioner. 
I’m going to go with “no”. The reason is that there’s nothing analogous to inertia in heat flow. Mathematically, the difference is that heat flow is described by a first order equation, whereas the equations of motion are second order.
The “air conditioner” seems to me to be more analogous to an elevator lifting someone. And even then, the analogy breaks down if the elevator is moving upwards very fast.
>The reason is that there’s nothing analogous to inertia in heat flow.
Yes, and this actually approaches a far deeper and mysterious thing. In the study of dynamical systems, with through-variables and cross-variables, and what I think of as their mirror images of one another in different orders of time differentiation, most such dynamic systems have analogous components. Thus, in electricity you have voltage and current and resistance and inductance and capacitance, and in mechanism you have (I hope I’m lining these up right, as it’s not trivial) displacement and force and inertia and springs, and in fluid mechanics you have flow and pressure and tanks. All these things work together in ways that you have to study closely to, for example, create a nice regulation system, or damp out oscillations optimally.
The weird thing is that there is nothing corresponding to inertia or inductance in the dynamics of thermal systems, and (if I remember right) experts have no idea why. Of course, Fourier’s equations state the system has no such term; what is significant here is why that would have been the case.