Personally, I would not refer to time as measured by a clock as "physical time". I would call it "clock time" or some such. Time as measured by a clock is stochastic. It only follows from the second law of thermodynamics, which is not fundamental law. It doesn't even make an assertion that is true in all possible worlds that have the same fundamental law as ours. And that have a Bug Bang.

There is a possible world where my fair coin has always come up heads, even though I have tossed it a quadrillion times. Likewise, there is a possible world in which my pocket watch has always run backwards, without being broken in any way. — Douglas Alan

The second law of thermodynamics gives a preferred overall direction for time, but the rate of time is established by any number of regular physical processes, such as the vibrations of a cesium atom, which are used for the standard atomic clock. It is the availability of such physical processes - plus an overall direction - that gives us time as we normally understand it in physical sciences. (A direction is still necessary to provide an order to the cycles of a physical process - otherwise you can't really say that this cycle occurred earlier or later than that cycle.) We can also talk about subjective psychological time and other kinds of time - these I distinguish from what I call physical time (or clock time, if you prefer). The coordinate time of the Minkowski or the Lorentzian manifold is one of the species of time that, I argue, is not identical to this generalized concept of physical time.

The second law of thermodynamics may not be fundamental, in the sense that it is reducible to physics at a lower scale, but all you need for it to obtain is a non-equilibrium state - and then it becomes as inevitable as any fundamental law. Almost as inevitable. I take your point about it being statistical. But does this strike a blow against the idea that thermodynamics is essential to our understanding of time? I understand probability and statistics epistemically, so if you tell me that the probability of an outcome is 1 - 10^-45 (that's the probability of a quadrillion coin tosses not all coming up heads), that's probably better than the confidence level of all our fundamental physics experiments combined.

The fact that clocks are not reliable in all possible worlds, or after the heat death of the universe leads me to exactly the opposite conclusion: We should not be making any profound metaphysical conclusions at all based on clock time. We should stick to relativistic time. — Douglas Alan

I am with you, to a point: I am not big on metaphysics. I would rather confine myself to more modest phenomenological models.

Note that when you object to thermodynamics playing any part in the definition of time on the grounds that other possible worlds with the same fundamental laws may not exhibit such thermodynamic asymmetry as is observed in our universe, you are already deep into metaphysical theorizing, perhaps without even realizing it.

relativistic time is not identical to physical time (the time that physical clocks measure) — SophistiCat

Actually, you never really defined what you mean by 'relativistic time'. You say physical time is that which clocks measure. I think physics would say that a clock measures proper time, a frame independent property of any timelike worldline. Clocks, not being confined to a single point, cannot be perfect, just like the exact length of a curving road is ambiguous because the road has nonzero width.
I digress. What do you mean by relativistic time? You said the two are usually the same except after heat death, but that doesn't tell me what you mean by the term, especially since a clock cannot exist in heat death conditions. It seems that time in general (both duration and direction) fade to meaninglessness along with most other physical concepts.

Actually, you never really defined what you mean by 'relativistic time'. You say physical time is that which clocks measure. I think physics would say that a clock measures proper time, a frame independent property of any timelike worldline. — noAxioms

Sure. But as you note later, proper time can be well-defined even in the absence of anything to mark its passage, and that is where I see a problem. You could have a relativistic spacetime with nothing else, and formally it would still have time of every description. But would it be physically meaningful?

Of course, here we run into the more general question of the nature of the physical law. Are laws ontologically prior to things and events that are subject to them? Do they describe potentialities (like what would happen if that empty spacetime did have something in it to demonstrate its relativistic properties)? Or do laws serve only to connect the dots, describe the regularities in our observations? In the latter case - and only in the latter case - it would be unjustified to talk about time in the absence of anything that is time-like. I think I lean towards the Humean regularity view.

One might object that clock time, as instantiated by physical clocks (regular processes), is a naive concept and that relativistic time is more fundamental. But clocks are epistemically prior to our concept of time. The reason we came up with the theory of relativity in the first place was to give a better account of clock-based time (among other things). Without clocks what is the point of relativity?

An empty, flat spacetime with a positive cosmological constant is an interesting case. Formally, there is a temporal process (expansion of spacetime), but with nothing clock-like to register the passage of time, does it still make sense to talk about time? Does it even make sense to model it with a Lorentzian manifold and Einstein's field equations?

Near as I can tell, you mean something like 'coordinate time' when using the phrase 'relativistic time'. Coordinate time and physical (proper) time is the same thing for an inertial worldline with the time axis of the coordinate system aligned with said worldline.

Sure. But as you note later, proper time can be well-defined even in the absence of anything to mark its passage, and that is where I see a problem. You could have a relativistic spacetime with nothing else, and formally it would still have time of every description. But would it be physically meaningful? — SophistiCat

No, time without change is meaningless, as is say motion of an object in the absence of other objects. The words might as well be invisible pink unicorns.

Of course, here we run into the more general question of the nature of the physical law. Are laws ontologically prior to things and events that are subject to them? Do they describe potentialities (like what would happen if that empty spacetime did have something in it to demonstrate its relativistic properties)? Or do laws serve only to connect the dots, describe the regularities in our observations? In the latter case - and only in the latter case - it would be unjustified to talk about time in the absence of anything that is time-like. I think I lean towards the Humean regularity view.

I seem to be in the latter camp. Spacetime with nothing in it doesn't have relativistic properties.

The reason we came up with the theory of relativity in the first place was to give a better account of clock-based time (among other things). Without clocks what is the point of relativity?

Can there be time without clocks? Surely there are primitive people and animals with an awareness of time, but they also arguably have clocks, however inaccurate. If the speed of light was a lot slower, its properties could be more apparent and intuitive to primitive beings who may not have developed accurate measurements for it yet. Similarly, our intuitive perception of time as flowing is only there because that perception makes us more fit, not because time necessarily flows.

An empty, flat spacetime with a positive cosmological constant is an interesting case.

Meaningless. If it has stuff, I don't think it can be flat. If it were, an inertial frame could foliate the space, and it cannot.

Near as I can tell, you mean something like 'coordinate time' when using the phrase 'relativistic time'. — noAxioms

Time is an integral part of relativistic spacetime. If you need a specific measurement of time, you can use proper time in conjunction with a specific timelike worldline, or a coordinate time in conjunction with a specific reference frame - doesn't matter, since they are all mutually convertible.

Can there be time without clocks? — noAxioms

I mean "clocks" in a general sense, as any observable regular physical process, such as an oscillation.

I don't think relativity requires a (semi) block universe. One physicist once declared us to be climbing along our worldlines, experiencing life on the way. This supposes the worldlines laid out for the material world to climb into. Like an intricate network of iron rails guiding the paths of particles into increasingly ordered structures while new stretches of rails spring off to harbor increasing entropy. The initial rail at the big bang is a closed rail guiding particles along to circle in spacetime (virtual particles), connected to a previous railwork of a previous big bang. Why don't the particles move in the opposite direction on the rail systems?

Can there be time without clocks? — noAxioms

Yes. Clocks measure time. Put a clock next to an irreversible process (which are all real particle processes) and you can measure how many periods it took. Which always is an approximation, as perfectly periodic motion is reversible and such process doesn't exist, the exception being the motion of virtual particles, being the only presence in the 3D space of the pre-inflationary era, "waiting" to be set in thermodynamic unidirectional motion before the sign is given by two previous 3D universes accelerating away from the 4D circularity.

The time used in relativity is an ideal, reversible clock, of which you can't tell it's going forwards or backwards.

Eternalism has a long history.

Jesus said to them, “Truly, truly, I say to you, before Abraham was, I am.” — John8:58

The searchlight in the block universe that is everywhere at once, is consciousness, which is nothing other than God's knowledge. But it is surely hubris for ordinary mortals to pretend to this god's eye view. Therefore I am a presentist - it's all "I" can know.

Well, if the universe is a block of 4D spacetime, the so-called now is a slice of it. Depending on the angle of that slice, I could be coevals with Socrates or Charles Darwin or Werner Heisenberg or (even) Lucy the hominin or dinosaurs or you get the idea!

Well, if the universe is a block of 4D spacetime, the so-called now is a slice of it. Depending on the angle of that slice, I could be coevals with Socrates or Charles Darwin or Werner Heisenberg or (even) Lucy the hominin or dinosaurs or you get the idea! — Agent Smith

And what or who determines the motion inside the block?

And what or who determines the motion inside the block? — Haglund

Frankly, your guess is as good as mine!

Well, if the universe is a block of 4D spacetime, the so-called now is a slice of it. Depending on the angle of that slice, I could be coevals with Socrates or Charles Darwin or Werner Heisenberg or (even) Lucy the hominin or dinosaurs or you get the idea! — Agent Smith

Get the angle right and we could see T. Rex chasing down a triceratops or, get this, even see the dino-killer asteroid streaking through the sky. 10-9-8-7-6-5-4-3-2-1-0!!! Kaboom! The end of the thunder lizard epoch!