What is the relationship between how time is viewed in thermodynamics and how time is viewed in general relativity? From my limited understanding of physics, it seems that the second law of thermodynamics, in which entropy never decreases over time in a closed system, relates to how time can only go forward i.e. the arrow of time. Additionally, general relativity highlights that space, time, and gravity are deeply intertwined i.e. gravity is curvature of space-time. What is unclear to me is how these different perspectives on time are related, hence the question. 
 A: Time for us was defined by the natural clocks of the seasons and the day night repetitions. If you look at the history of  time even the the units were variable in some civilizations.

In classical Greek and Roman times they used twelve hours from sunrise to sunset; but since summer days and winter nights are longer than winter days and summer nights, the lengths of the hours varied throughout the year.
Hours did not have a fixed length until the Greeks decided they needed such a system for theoretical calculations. Hipparchus proposed dividing the day equally into 24 hours which came to be known as equinoctial hours. They are based on 12 hours of daylight and 12 hours of darkness on the days of the Equinoxes. However, ordinary people continued to use seasonally varying hours for a long time. Only with the advent of mechanical clocks in Europe in the 14th Century, did the system we use today become commonly accepted.

These units were locally  in use in the middle east afaik until the beginningof last century, where night and day were divided by twelve between sunset and sunrise. They are still used for religious purposes afaik.
There is a history of mechanical clocks, and then came the atomic and quantum mechanical revolution and the atomic clocks which can define the unit of time within any system having that atom.
In parallel, physics defined the arrow of time thermodynamically, by the change in entropy , and that is the arrow of time we use on the axis of time to define a $+$ and
a $-$ .
We transfer this arrow of time to special relativity, and have an enormous number of data to know that it works without mathematical contradictions, and to general relativity. We are creatures dominated by classical thermodynamics. In science fiction one might find other frames of reference, but that is the fact, that our "coordinate" base is in classical dimensions.
You ask:

What is unclear to me is how these different perspectives on time are related, hence the question

Even for the  $(x,y,z)$  coordinates special and general relativity introduce distortions with respect to different kinematic frames, but they do not change the meaning of $+$ and $-$ for a relativistic transformation. It is just that our existence works one way by thermodynamic laws that the $-$ of time is inaccessible and an arrow has to be defined for the history of the universe.
There are no different perspectives, but expanded views keeping the same definition for the arrow of time from classical mechanics to relativistic, where coordinates acquire different behaviors, but the arrow of time is fixed by the fact of our physical existence in the classical realm.
A: In thermodynamics, the way time is viewed is connected in some ways to entropy. The second law of thermodynamics states that the entropy of an isolated system never decreases.
Such systems spontaneously evolve towards thermal equilibrium. This will give an arrow of time in the view of thermodynamics. 
For us, it is natural to look at macro objects to evolve chemically the way they do, because we are used to it. But the reality is that this second law of thermodynamics is what dictates the evolution of the macro systems that we observe in our life. Chemical reactions in a macro system obey the second law of thermodynamics, and so we tend to be used to viewing these macro systems to chemically evolve the way they do. This thermodynamical and chemical law will give an arrow to time in our lives. We are used to it, because that is how everything we view evolves in the macro world.
Now GR will explain you that if you place two clocks in two different gravitational zones, they (as the gravitational potential will be different at the two clocks' position), the clocks will tick differently when viewed from afar.
If you look at your own clock locally, it will always tick normally. It is when you compare your clock to another clock that is in another gravitational field (that has a different gravitational potential) when you will see that the clocks tick at different rates. A very good way to learn about this is the Shapiro delay.
Now you are asking how these different ways of viewing time are connected. You are basically asking whether there is casuality between GR time dilation and the laws of thermodynamics.
Let's take a very strange example. 


*

*Let's assume that GR time dilation causes the laws of thermodynamics. 


This would mean that the gravitational potential, and the difference between the gravitational potential (between different places in space) causes the laws of thermodynamics. Let's say that the earth is an isolated system and it is moving towards thermal equilibrium. Is this second law of thermodynamics caused by GR time dilation? Is this caused by the fact that the gravitational potential will dictate how fast time will flow on earth (compared to another place in space where the gravitational potential is different)? No it does not.
The laws of thermodynamics are based on QM processes. There is no way how we could say that entropy is caused by GR time dilation.


*Now let's try the opposite, let's say that GR time dilation is caused by the laws of thermodynamics. 


This would mean that there is a way we can prove that, the laws of thermodynamics, which is based on QM processes, cause GR time dilation. So basically QM processes cause the gravitational potential, and the difference between the gravitational potential to cause time to flow at different rates at different points in space (where the gravitational potential is different). To prove this, we would need to prove that the gravitational potential is determined by QM processes. This would be quantum gravity. As of today, we have no accepted theory of everything (TOE), that would include GR and QM, and that could explain both on the QM level. There are theories like string theory, that could be like that, but they need to be experimentally tested and proven.
To answer your question, to say (or even to think about) that the gravitational potential (and gravitational effects) is caused by the same QM processes that cause the laws of thermodynamics, we would need first of all a QM level description of GR, and a TOE.
