quantum field theory [@Weinberg]. In the case of gravity we deal with space-time symmetry, and we should take into account explicit dependence of the Lagrangian and the measure on space-time coordinates. One can check that the sum of gauge-fixing and ghost parts of the action (\[gfix-act1\]), (\[ghost-act1\]) is not invariant under transformations (\[delN1\]) – (\[coord.tr\]). In some works the BRST invariance is guaranteed by asymptotical boundary conditions for ghosts and Lagrange multipliers [@Hennaux; @Hall]. The legitimacy of asymptotic boundary conditions is questionable in the case of space-time of arbitrary topology. Therefore, we seek for a BRST invariant form of the action without appealing to any additional conditions. One can check that to ensure its BRST invariance we have to add to the action the following term containing only full derivatives and not affecting the set of equations obtained in Section 2: $$\begin{aligned}
S_{(add)}&=&\int dt\int\limits_0^{\infty}dr
   \left(\frac d{dt}\left[\bar\theta_0\left(\dot N-\frac{\partial f}{\partial V}\dot V
     -\frac{\partial f}{\partial W}\dot W\right)\theta^0\right]
    +\frac d{dr}\left[\bar\theta_0\left(\dot N-\frac{\partial f}{\partial V}\dot V
     -\frac{\partial f}{\partial W}\dot W\right)\theta^r\right]\right.\nonumber\\
\label{add-act}
&+&\left.\frac d{dt}\left[\bar\theta_r\left(\dot N^r-\frac{\partial f^r}{\partial V}\dot V
     -\frac{\partial f^r}{\partial W}\dot W\right)\theta^0\right]
    +\frac d{dr}\left[\bar\theta_r\left(\dot N^r-\frac{\partial f^r}{\partial V}\dot V
     -\frac{\partial f^r}{\partial W}\dot W\right)\theta^r\right]\right).\end{aligned}$$

As was mentioned in Section 2, the gravitational part of the action in (\[full-act2\]) is 