For an ideal gas in a 3d box, the "equation of state" that relates P(ressure), T(emperature), and V(olume) is PV = nRT, where R is a constant and n is the number of particles. You can rewrite this as:
P = nRT/V
Compressing the gas (by shrinking V) while keeping n constant will result in a higher pressure P. You can imagine a cylindrical piston, where the pressure is the force per unit area it takes to hold the piston down.
The important thing to remember is that this equation describes an ideal gas in a 3d box. An ideal gas is made of tiny particles that only interact with the piston's walls, but not with each other. This is an approximation to reality, but a decent one when the gas is low density.
Weird things can happen when particles interact, and when the dimensions are different. The equation of state above is one example of how P, V, and T can be related, but if the particles interact a little more the equation of state can change. And if they interact a lot more (by, say, becoming a liquid) the equation of state can change again.
This paper is talking about photons in a 2d box (an optical trap), and in part talks about measuring/confirming its equation of state -- the relationship between P, T, and V. These particles have a peculiar kind of interaction, where the photons don't really interact unless they're in a special state. I'm gonna quote the relevant part of the paper (from https://arxiv.org/pdf/2112.12787.pdf):
> It is well understood that as the thermal wave packets spatially overlap the
classically expected decrease in compressibility with density (it is harder to compress a dense gas than a dilute one) is replaced by a compressibility increase stemming from the quantum-statistical occupation of low-lying
energy levels, reducing the energy cost for compression as compared to the classical gas case. In the extreme high-density limit of an infinite-size deeply degenerate gas, bosons can be added to the system at essentially vanishing energy cost...
My translation: it's normally harder to compress a normal gas the more you squeeze it, but for a photon gas it's different. Because photons are bosons, as you compress them (or cool them), they tend to group together in a special configuration. That special configuration is called a Bose-Einstein Condensate (BEC). In a BEC, a meaningful fraction photons pile into the ground state. (This is what the paper calls "degeneracy" -- quantum particles being in the same energy state.)
(According to the paper this is NOT possible when the 2d configuration is "infinite", but does happen in some cases when the trap is finite, as is the case for a real experiment.)
To say more than this would be tricky and take an expert, which I am not. But I think this might illuminate some of the subtler aspects of the experiment which may take away some of the uneasiness that you're feeling.
But I still think the uneasy feeling is justified: when you get into quantum thermodynamics some things become a little trickier to reason about, as intuitions about pressure, volume, and temperature begin to break down somewhat.
Edit: as a final clarification, I think they try to keep the number of photons constant:
> To maintain a steady-state photon number inside the cavity, continuous pumping is required to compensate
losses from mirror transmission.
A reasonable intuition might be: as the number of photons in the ground state (N_0) increases, the remaining photons that can provide significant pressure (N) reduces. (I am unsure about this, because I don't know how much pressure ground-state photons contribute.)