$M$ is a compact Riemannian manifold,$g_{ij}$ and $f(t)$ is defined as first picture.
I want to compute (as equality with red line in second picture) $$ \int_M-\nabla_if\nabla_i(2\Delta f-|\nabla f|^2)e^{-f}dV \\ =\int_M-\nabla_if(2\nabla_j(\nabla_i\nabla_jf)-2R_{ij}\nabla_jf-2\langle \nabla f ,\nabla_i\nabla f\rangle)e^{-f}dV $$
I try that: $$ \nabla_i(2\Delta f-|\nabla f|^2)= (2\nabla_i(g^{kl}\nabla_k\nabla_lf)-\nabla_i\langle \nabla f,\nabla f\rangle) $$ I assume that, although $g_{ij}(t)$ evolve with $t$, but when $t$ is given, I can choose a normal coordinate such that $g_{ij}=\delta_{ij}$. Then $\nabla_kg_{ij}=0,\Gamma_{ij}^k=0$. Then: $$ \nabla_i(2\Delta f-|\nabla f|^2)= (2\nabla_i(g^{kl}\nabla_k\nabla_lf)-\nabla_i\langle \nabla f,\nabla f\rangle)\\ =2\nabla_i(\nabla_j(\nabla_jf))-2\langle \nabla_i\nabla f,\nabla f\rangle \\ =2\nabla_j(\nabla_i(\nabla_jf))-2\langle \nabla_i\nabla f,\nabla f\rangle $$ But there is not $-2R_{ij}\nabla_jf$.
The above picture is from 201th page of this paper.