As mentioned by Wilie Wong, I modified to the following verison:

Let $M$ be a closed smooth $4$ manifold.

Q Suppose that $c>0$ is any positive number, can we find a Riemannian metric $g$ on $M$, such that the $\int_MScal^2_gdv_g=c$, where $Scal_g$ denotes the scalar curvature of $g$? If not, for any small $\epsilon>0$, can we find a metric $g_\epsilon$ such that $|\int_MScal^2_{g_\epsilon}dv_{g_\epsilon}-c|<\epsilon$?

PS I do not know whether the question is trivial or not. Any reference is welcome.

As mentioned by Willie Wong, I modified to the following version:

Let $M$ be a closed smooth $4$ manifold.

Q Suppose that $c>0$ is any positive number, can we find a Riemannian metric $g$ on $M$, such that the $\int_MScal^2_gdv_g=c$, where $Scal_g$ denotes the scalar curvature of $g$? If not, for any small $\epsilon>0$, can we find a metric $g_\epsilon$ such that $|\int_MScal^2_{g_\epsilon}dv_{g_\epsilon}-c|<\epsilon$?

PS I do not know whether the question is trivial or not. Any reference is welcome.

As mentioned by Wilie Wong, I modified to the following verison:

Let $M$ be a closed smooth $4$ manifold.

Q Suppose that $c>0$ is any positive number, can we find a Riemannian metric $g$ on $M$, such that the $\int_MScal^2_gdv_g=c$, where $Scal_g$ denotes the scalar curvature of $g$? If not, for any small $\epsilon>0$, can we find a metric $g_\epsilon$ such that $|\int_MScal^2_{g_\epsilon}dv_{g_\epsilon}-c|<\epsilon$?

PS I do not know whether the question is trivial or solved in some case(e.g. Kahler manifold with fixed Kahler class). Any reference is welcome.

As mentioned by Wilie Wong, I modified to the following verison:

Let $M$ be a closed smooth $4$ manifold.

Q Suppose that $c>0$ is any positive number, can we find a Riemannian metric $g$ on $M$, such that the $\int_MScal^2_gdv_g=c$, where $Scal_g$ denotes the scalar curvature of $g$? If not, for any small $\epsilon>0$, can we find a metric $g_\epsilon$ such that $|\int_MScal^2_{g_\epsilon}dv_{g_\epsilon}-c|<\epsilon$?

PS I do not know whether the question is trivial or not. Any reference is welcome.