  It is well known in quantum field theory that the off-shell effective action
depends on the gauge choice and field parametrization used in calculating it.
Nevertheless, the typical scheme in which the scenario of asymptotically safe
gravity is investigated is an off-shell version of the functional
renormalization group equation. Working with the Einstein-Hilbert truncation as
a test bed, we develop a new scheme for the analysis of asymptotically safe
gravity in which the on-shell part of the effective action is singled out and
we show that the beta function for the essential coupling has no explicit
gauge-dependence. In order to reach our goal, we introduce several technical
novelties, including a different decomposition of the metric fluctuations, a
new implementation of the ghost sector, and a new cut-off scheme. We find a
non-trivial fixed point, with a value of the cosmological constant which is
independent of the gauge-fixing parameters.
