variation of the action with respect to $g_{\mu\nu}$, leading to $$\begin{aligned}
\label{dyn}
G_{\mu\nu}=T_{\mu\nu}+\tilde{T}_{\mu\nu},
$$ where $G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R$ is the well-known Einstein tensor, while the tensor $\tilde{T}_{\mu,\nu}$, coming from the term $f(\chi)$, is given by $$\begin{aligned}
\tilde{T}_{\mu\nu}
\equiv
\left(f-\chi f_{\chi}+u^{\alpha}\chi_{,\alpha}f_{\chi\chi} 
\right)g_{\mu\nu}-f_{\chi\chi}(u_{\nu}\chi_{,\mu}+u_{\mu}\chi_{,\nu}
+u^{\alpha}\chi_{,\alpha}u_{\mu}u_{\nu}).\end{aligned}$$

On the other hand, the variation of the action with respect to the scalar field $\phi$ leads to the following conservation equation, $$\begin{aligned}
\label{conser}
-\Box \phi+ V_{\phi}-\nabla_{\mu}\left(  \frac{1}{\sqrt{-\phi^{,\alpha}\phi_{,\alpha} }} \left( \partial^{\mu}f_{\chi}+u^{\mu}u_{\alpha}\partial^{\alpha}f_{\chi}   \right) \right)=0,\end{aligned}$$ which differs from the usual one $-\Box \phi+ V_{\phi}=0$. However, for the FLRW geometry it leads to the standard conservation equation, $$\begin{aligned}
\label{conserb}
\ddot{\phi}+3H\dot{\phi}+V_{\phi}=0.\end{aligned}$$

The background
--------------

Considering backgrounds (i.e., solutions of (\[conserb\])) satisfying $\dot{\bar{\phi}}(t)>0$ all the time, the simplest way to obtain the dynamical equations is as follows: We consider the flat metric $ds^2=-N^2(t)dt^2+a^2(t)\delta_{ij}dx^idx^j$, which leads to $\chi=\frac{3H}{N}$, where $N(t)$ is the lapse function. Hence, after integration by parts, the action becomes $$\begin{aligned}
S={\mathcal V}\int a^3N\left(-\frac{3H^2}{N^2}+ \bar{f}\left(\frac{3H}{N}  \right)+{\mathcal L}_{matt}         \right) dt,\end{aligned}$$ where ${\mathcal V}$ is the volume of the fixed elementary spatial cell, where all spatial integrations are performed, and where the matter Lagrangian simplifies to $$\begin{aligned}
\label{Lagrangian}
{\mathcal L}_{matt} =\frac{{\dot{\bar{\phi}}}^2}{2N^2}-V(\bar\phi).\end{aligned}$$

Performing the variation with respect to $N$ and taking at the end $N=1$, one obtains 