There are some discrepancies on how to introduce entropy in classical kinetic theory. In what follows $f(r,p,t)$ is the usual one-particle distribution function of a monatomic gas, normalised to the number of particles $N$.
The negative of Boltzmann's $H$ function is often taken as the entropy:
$$ S = - k_B \int f \ln f \, d^3 r \, d^3 p, \quad N = \int f \, d^3 r \, d^3 p. $$
On the other hand, in Landau and Lifshitz it is $$ S = k_B \int f \ln \frac{e}{f} \, d^3r \, d^3 p, \quad N = \int f \, d^3 r \, d^3 p $$ This is actually derived and I remember to follow the derivation and find it completely satisfactory, though I forgot it by now.
Another book, "Beyond equilibrium thermodynamics" by Hans Öttinger, arrives via the projection-operator technique to
$$ S = - k_B \int f \ln \frac{f}{N} \, d^3r \, d^3 p, \quad N = \int f \, d^3 r \, d^3 p. $$
All of these expressions differ by a constant depending on $N$. I wonder what would be the most general/universal expression for entropy as a functional of a one-particle distribution function, that would be valid even outside the field of classical kinetic theory. For example, I suppose it would be nice to reproduce the Sackur–Tetrode formula in equilibrium: $$ S = k_B N \left(\frac{5}{2} + \ln \left[\frac{V}{N} \, \biggl(\frac{2\pi \, m \, k_B T}{h^2}\biggr)^{3/2} \right] \right). $$