The Hamiltonian formulation of general relativity - either in the ADM formalism or in Ashtekar variables - is straightforwardly a gauge theory. While the BRST formalism has primarily been developed to quantize such theories, it can be applied to such theories without the quantization step, in the sense that we identify the (classical) observables as the algebra of BRST-invariant function(al)s of the dynamical variables.
The BRST formalism has been applied to the Ashtekar formulation in "Brst Cohomology and Invariants of 4D Gravity in Ashtekar Variables" by Chang and Soo and "BRST cohomology of Yang-Mills gauge fields in the presence of gravity in Ashtekar variables" by Schweda et al. to find certain BRST-invariants of GR (coupled to a Yang-Mills field in the latter paper) using Wess-Zumino descent, essentially confirming the "usual" set of invariants (e.g. cosmological constant, E-H action, action for topological gravity).
However, I have not found any direct continuation of this line of reasoning. The main question is:
What is the full algebra of BRST-invariant observables (including their Poisson brackets)? Are the invariants computed in these papers all the possible ones up to trivial combinations, or just the ones that are "easy" to find?
As for why this question is relevant: If this algebra is known, then a follow-up question would be: What, if anything, obstructs quantization of this system? Since all the resulting invariant operators with zero ghost number are not local, but integral, the usual logic of perturbative QFT does not seem to apply (and in particular the standard argument about non-renormalizability of linearized gravity around a Minkowski background does not transfer immediately).
While loop quantum gravity uses the Hamiltonian formulation of Ashtekar variables as its starting point in a similar way, it does not seem to actually attempt to implement the BRST procedure, so unless my impression of this is mistaken, it does not contain the answer to this question.
