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.
