In Weinberg's paper on the cosmological constant problem (CCP), he states that diffeomorphism invariance is always broken by the presence of any given metric $g_{\mu\nu}$. He then goes on to say that there is a residual $GL(4)$ symmetry remaining if we choose the metric (and the set of scalar fields introduced as a putative solution to the CCP) to be constant.

I'm quite confused by this though. In particular, I don't see why the choice of metric breaks diff. invariance? I thought the whole point was that, if $g_{\mu\nu}$ is a solution to the Einstein field equations, then any other metric $\hat{g}_{\mu\nu}$ related to $g_{\mu\nu}$ by a diffeomorphism, i.e. $\hat{g}_{\mu\nu}=(\phi^{\ast}g)_{\mu\nu}$, is also a solution. What am I missing here?

Also, why is there a residual $GL(4)$ symmetry? Is it precisely because the fields are constant, i.e. the on-shell solutions are translationally invariant?

In Weinberg's paper on the cosmological constant problem (CCP), he states that diffeomorphism invariance is always broken by the presence of any given metric $g_{\mu\nu}$. He then goes on to say that there is a residual $GL(4)$ symmetry remaining if we choose the metric (and the set of scalar fields introduced as a putative solution to the CCP) to be constant. (For clarity, he discusses these things on page 4, left-hand column, 2nd paragraph onwards).

I'm quite confused by this though. In particular, I don't see why the choice of metric breaks diff. invariance? I thought the whole point was that, if $g_{\mu\nu}$ is a solution to the Einstein field equations, then any other metric $\hat{g}_{\mu\nu}$ related to $g_{\mu\nu}$ by a diffeomorphism, i.e. $\hat{g}_{\mu\nu}=(\phi^{\ast}g)_{\mu\nu}$, is also a solution. What am I missing here?

Also, why is there a residual $GL(4)$ symmetry? Is it precisely because the fields are constant, i.e. the on-shell solutions are translationally invariant?