From the idea that the constant that equates to dark energy in the stress energy tensor is not commonly thought to be a constant. People didn't call it that way as some joke. This isn't even up to astronomers, this is something that mathematicians will tell you. And it could potentially cause some real trouble not just for our models of the universe but for the basic mathematics of relativity if it is changing. Motivating that alone will we hard, not to speak of the consequences down the line.
> it could potentially cause some real trouble not just for our models of the universe but for the basic mathematics of relativity if it is changing.
The Einstein Field Equation can accommodate models in which the density of dark energy changes with time just fine. The paper mr_mitm linked to discusses such models. There is no "real trouble" for the basic math of relativity involved here at all.
You can do all sorts of stuff with the Einstein field equations (or any differential equation for that matter). People do this all the time because it's an easy paper. Many people for example make the Ricci scalar into a function. Aka "what happens if I magically introduce this." That way you can sometimes explain some stuff on one end while breaking stuff at other ends. But whenever you introduce such new degrees of freedom, you need to accommodate them somewhere in the overall picture. f(R) gravity in that sense would be better motivated, because in the infrared it boils down to a dumb extra scalar field. And you could get tons of those easily from string theory. But if you want to use local field theories of gravity to explain horizons, then you need something like holography to provide a consistent picture. Even if you go for metastable degenerate vacua, it will be really hard to explain how this leads to an apparently continuously varying cosmological constant. Is it possible? Maybe. Is it more likely that we misunderstood stellar evolution? Absolutely.
No one is saying the cosmological constant can vary. The idea is simply that if dark energy is dynamical, it clearly cannot be explained by a cosmological constant, but perhaps a scalar field instead. And my point was that this idea is nothing new, unlike the headline seems to suggest.
As explained above this scalar field would manifest differently in the equations and basically be something that enters alongside a cosmological constant. Making dark energy or Λ itself vary at the level of the Einstein field equations means you're essentially making an integral constant vary which is not straightforward as people here purport.
> this scalar field would manifest differently in the equations
It shows up in the Einstein Field Equation as part of the stress-energy tensor. That is not a problem at all.
> and basically be something that enters alongside a cosmological constant.
The "cosmological constant" can just as easily be moved to the RHS of the Einstein Field Equation and also considered part of the stress-energy tensor, yes. There is no issue with that at all.
> Making dark energy or Λ itself vary
Is something nobody has claimed to be doing, as the GP said, so you are attacking a straw man.
"Varying dark energy density" does not mean trying to make the cosmological constant vary. It just means adding some other component of stress-energy whose density does not have to be constant, but which still produces accelerated expansion. Anything with pressure less than minus 1/3 of its energy density will do that. A scalar field is the simplest such thing, but not the only possibility. None of this poses the slightest problem for the Einstein Field Equation.
From the idea that the constant that equates to dark energy in the stress energy tensor is not commonly thought to be a constant. People didn't call it that way as some joke. This isn't even up to astronomers, this is something that mathematicians will tell you. And it could potentially cause some real trouble not just for our models of the universe but for the basic mathematics of relativity if it is changing. Motivating that alone will we hard, not to speak of the consequences down the line.