Entropy works in the same Boltzmannian way - within a shell (a Gaussian surface) one takes a measure of the number of configurations of the enclosed elements which produce the same observables outside the shell.

That is, one takes a coarse-grained macrostate "a living human in a functioning spacesuit", a volume (say a cubic metre), and a fine-grained microstate (say a cubic millimetre); the more pairs of microstates one can swap without changing the macrostate, the more the entropy. If you swap part of the living human's aorta with shards of helmet, toenail, or vacuum, you quickly get a non-living, therefore non-metabolizing, therefore observably cooling human, so the entropy is well below a maximum.

But if our macrostate is "a cubic metre of vacuum" you can swap any sized microstate around and still get all the same observables as the original "cubic metre of vacuum" -- entropy is therefore maximal for the shell around that volume. We can repeat this procedure for arbitrarily-sized Gaussian surfaces, and arbitrarily fine microstates.

We recover the second law of thermodynamics by observing that wherever in the entire cosmos you place your shell, you are more likely to enclose a high entropy region than a low entropy one. In an expanding and diluting universe like ours, where more and more high quality vacuum appears between galaxy clusters, there are more places where one's shell will enclose a very high entropy region than a very low one. We then can consider Boltzmann's view of the second law of thermodynamics as it being infinitely improbable to have a completely dynamically ordered state.

Let's contrast the Janus point with a Lemaître style regression cosmology.

In the latter we simply look at the expansion history in reverse, and extrapolate through ever denser and hotter and lower-entropy configurations and (following classical General Relativity) end up at an inevitable singularity. This has some problems, mainly that nobody knows how matter works at the much more extreme heats and densities than we can hope to produce in laboratory conditions on Earth, nobody is happy with a singularity because there is no way to predict that an actual singularity will decay into the fields of the Standard Model, and because the singularity contains everything, there is nothing outside the Gaussian surface to make observations of the macrostate. Returning to Boltzmann, the singularity itself must be completely dynamically ordered, because when it breaks down, it must be able to produce dynamical systems like galaxy clusters and cats.

However, if we somehow prevent the singularity, we might be able to make that prediction in principle even if we cannot do so now. We substitute the infinitely dense zero entropy singularity for merely extremely dense and extremely low entropy, and can at least in principle evade all of the problems in the previous paragraph. The Janus argument is that a singularity-free entropy minimum is plausible if shared with two regions with much higher entropy everywhere else. A shell around the entropy minimum has much less empty space in it than a shell around anywhere else in the two regions, so we can show this relatively low entropy by doing the swapping procedure above.

In both cases we make the argument that we can take a values surface at the entropy minimum and use dynamical laws to predict how that values surface will evolve. In the Lemaître-style system, we get a universe like ours; but in a system with a non-singular entropy minimum, we must have more than one region (one containing a universe like ours, one containing something else that evolved out of the entropy minimum).

In neither approach do we have the means to determine what the initial values should be, so abolishing the singularity Janus style doesn't seem to bring that much of practical calculational value. Moreover if we start with some late-time values surface on this side of the Janus point and work backwards, our known time-reversible dynamical laws do not lead to a Janus point but rather to a Lemaître-style singularity.

Let's return to recovering thermodynamics from the second-law discussion above.

The third law again comes from a statistical mechanics view. There is a unique low-energy state that is perfect vacuum. In an expanding universe, we have regions containing that state after it has been evacuated of galaxy clusters, dust, and gas, and the cosmic microwave radiation has become so sparse and cold as to essentially vanish (a bit technically, we can put in a comoving observer with a Eulerian view of the cosmic microwave background such that the characteristic wavelength of the CMB photons are longer than the observer's Hubble length). Those regions are nowhere near the entropy minimum in either the Janus or in the Lemaître configurations, but are nearly everywhere when sufficiently far in the future. (We somewhat circularly define the past as lowest entropy and future as highest entropy.)

At cosmological scales in an expanding universe there is no especially satisfying way to recover the first law of thermodynamics even though it is perfectly reasonable to treat the whole cosmos as the ultimate closed system. One can think of the expansion as an adiabatic and reversible process on the matter content, however, and that is part of the basis for the Lemaître and Janus models.

> I don't know how to make sense out of this

Well, me neither, frankly. Or rather, I can understand the goal of Barbour's thinking but I think it misses the point. We still have an extremely improbable configuration somewhere when the universe was much smaller and denser, and we have no way to recover that surface using observations made here-and-now. Worse, with what we know of gravitation -- specifically, if we accept Raychaudhuri's focusing theorem (which s a deep and interesting result of General Relativity) -- missing the focusing into a caustic is much less probable than focusing into a caustic. Once you have a caustic, you have a singularity, unless you have some magic means of avoiding it through unknown quantum effects. The Janus point doesn't even seem to open up that option, or rather, it appears to require either insanely good luck or quantum effects modifying General Relativity at energy scales which are astrophysical at modern times.

However, maybe just maybe what's on the other side of the Janus point has different physics that makes a Janus point (that produces our side with high probability) likely. And maybe just maybe in the far future when we can make truly enormous gravitational wave detectors we can spot gravitationally-lensed early-cosmos gravitational waves that might distinguish between a Janus-style early configuration and a singularity early configuration (or some other singularity-avoiding early configuration).

There are also some theoretical questions. The big one: what constrains the Janus point to having exactly two higher-entropy regions rooted to it? Also: are there false Janus points, i.e., is there a hierarchy of relatively low entropy configurations from which two+ higher-entropy regions sprout, but those regions still have enough entropy coupled with dynamical laws that (quoting the article) "The diameter will shrink to a minimum at some moment in time, then grow again"?

I think most working cosmologists would bet [a] a Janus configuration does not seem more probable than any other plausible early-universe configuration and [b] even if it were and thus abolished the singularity, it does not solve the vexing theoretical problems posed by the very early universe's extreme heat and density.

Finally, all of this has evaded Barbour's "timelessness" language, since that was largely missing from the Nautilus article, and my comments are instead rooted in the conventional concordance cosmology.