How is the entropy at the time of the Big Bang calculated? I've seen several reports that the entropy of the Universe at the time of the Big Bang was lower than it is now.
This seems impossible. 
If we view entropy as a function of the number of micro states consistent with a given macro state, I cannot imagine how the entropy of what was basically a giant fire (at the time of the Big Bang) is not higher than the entropy of the current state of the Universe, which has stable planets, and life.
This is not inconsistent with the Second Law as a local phenomenon, it just says what observation plainly suggests -
Gravity caused the universe to become less complex over time, eventually supporting life.
On a smaller scale, the entropy of a fire must be higher than the entropy of a stable object, since the fire is obviously rapidly changing state, and therefore, presumably, is capable of being in a larger of number of states, while still being a fire. In contrast, a cup probably does change states somewhat, but obviously not as often, and obviously not on the same scale, in that the macro state of a cup is roughly constant.
As a result, common sense suggests that the entropy of a fire is higher than the entropy of a stable object.
Also, the temperature of the universe has consistently decreased over time:
http://www.astro.ucla.edu/~wright/BBhistory.html
https://www.sciencedaily.com/releases/2013/01/130123101622.htm
So, why do physicists say that the entropy of the Universe around the time of the Big Bang was lower than it is today?
It doesn't make any sense, yet I've heard two reputable physicists make this claim, which is on its face almost certainly incorrect.
I'm wondering if someone can put forward a simple calculation that explains where this claim comes from, rather than simply quoting the Second Law, and saying it must be so, when available evidence strongly suggests otherwise.
This would involve taking the temperature at the time of the Big Bang, and expressing the entropy as a function of that temperature, and showing that entropy decreases as you move away from the Big Bang in time - and I don't see that happening.
The bottom line is that the universe is both colder and more structured than it once was. This means you can describe its structure with less information. So any statement that the universe has increased in entropy, or complexity, since the Big Bang, borders on nonsense, in my opinion.
It's also perfectly reasonable to assume the universe became more structured with time, since it started out so hot, that its kinetic energy overpowered any of the stabilizing forces like gravity, charge, etc. 
You can pretend otherwise, but the reality is that gravity decreases local complexity over time, and other small scale forces do the same, forming atoms, eventually molecules, and then apparently, life itself.
 A: Two issues: 
1) Your objection would seem to apply to any explosion. The initial state is hot, the final state is cold. What happens is that the initial state has high entropy density, but small volume. During the explosion volume increases, and entropy density decreases. Overall, the total entropy goes up somewhat, because of non-equilibrium processes (as required by the second law).
2) The main issue with the big bang is a different one. The entropy in non-gravitational degrees of freedom is large to begin with and does not increase by very much. However, for a system that interacts gravitationally the state of maximum entropy is a black hole. The initial state of the universe is believed to have had a fairly smooth geometry, so its entropy was much smaller than it could have been. 
A: The statistical mechanics definition of entropy for a system at thermodynamic equilibrium is

"a function of the number of micro states consistent with a given macro state"

In order to apply it to the Universe (does not matter if in its early stages or later) one should reply positively to a few preliminary questions


*

*is the universe a system in thermal equilibrium?

*do we know how a macro-state of the universe is characterized?

*is it conceivable to know what are the available micro-states of the universe at present time?


As far as I can see, answers to all these  questions are negative.
As a consequence, I cannot assign a verifiable meaning to any statement about the entropy of the Universe. I just take this kind of sentences as 
a loosely speaking by analogy. Nothing more than that.
A: I also don't understand why people sometimes say that the universe after the big bang was in a "low entropy" macrostate. (But, personally, I have never heard a physicist say this, so I don't think you should give the statement much weight.) According to the inflation model, the universe was all basically in thermal equilibrium right before inflation, which would suggest it was at maximum entropy.
The confounding factor here is indeed gravity. To my knowledge, there's no good way to ascribe entropy to a gravitation field that is not asymptotically flat, as is the case in an expanding universe. (How do you "count states," exactly?)
The reason that
 the universe cooled down since the big bang is that, due to general relativity, the universe has been expanding, separating all the hot stuff. Perhaps the entropy has not decreased, but the entropy density, entropy per unit volume, has decreased. This has allowed for life, etc. Obviously the universe is not in thermal equilibrium anymore.
(Also note that high entropy $\neq$ high temperature in general, these are different concepts.)
While I'm here, I'd like to address one misconception in your question:

On a smaller scale, the entropy of a fire must be higher than the
  entropy of a stable object, since the fire is obviously rapidly
  changing state, and therefore, presumably, is capable of being in a
  larger of number of states, while still being a fire. In contrast, a
  cup probably does change states somewhat, but obviously not as often,
  and obviously not on the same scale, in that the macro state of a cup
  is roughly constant.

The reason that a hot gas will have more entropy than a cold gas is that there are a larger range of velocities (actually, momenta) than the hot gas particles could be moving with: hence, more microstates. When there is more energy around, more states are "unlocked." In order to rigorously count microstates, which is a subject in an introductory statistical mechanics course, you must find the "phase space volume" of all the microstates, i.e. if you have $N$ particles, you must integrate
$$
\int_{\rm Vol} d^3 x_1 \ldots d^3 x_N d^3 p_1 \ldots d^3 p_N
$$
where $\rm Vol$ is the volume in phase space of your microstates, $x_i$ is the position of the $i^{\rm th}$ particle and $p_i$ is the momentum of the  $i^{\rm th}$ particle. That will give you your "number" of states, up to a dimensionful factor. The logarithm of the number of microstates is the entropy.
A: Judging by the various comments and answers I have seen posted so far, it seems to me that the main mistake made by the questioner is to assume that cold things always have lower entropy than hot things. This is wrong because the states counted in entropy measurements are states in position-momentum space (called phase space), not just momentum space. During an adiabatic expansion, for example, the entropy is constant but the temperature goes down.
Having said that, assessments of any property of "the whole universe" are always going to be metaphysically tricky. It is perhaps better to restrict our statements to identifiable parts of the universe, such as for example the part of it within the particle horizon. Assuming that the heat flow across the boundary of this region is small (and there is no reason for there to be much heat flow because the temperature is close to uniform), the entropy of that part has grown, for the same reason the entropy of any other isolated system grows under spontaneous processes.
The other interesting part of the picture is that large gravitating clouds are unstable against gravitational collapse, and in such a collapse some parts get hotter while other parts get colder. That can seem surprising when you first meet it, but in such processes the overall entropy is again increasing. After such collapse you get things like stars, and planets in orbit---and sometimes (well, at least once), life!
