Bohmian mechanics description of a large number of interacting atoms would require a large phase space due to the large number of classical degrees of freedom. The entropy per atom is given as the logarithm of the volume of the phase space of states that are accessible at thermal equilibrium. An atom's heat capacity is close to $k_B$, but Bohm's theory seems to be in conflict with this. If there is a way to compute an atom's heat capacity in Bohm's theory in a natural way that doesn't include some ad-hoc solution which will conflict with other kinds of physics experiments, I'm curious to see the calculation.
One should distinguish here de Broglie-Bohm theory for the general situation outside the equilibrium, and that for quantum equilibrium. Entropy is defined as usual by $H=-\int \rho \ln \rho dq$. Outside the quantum equilibrium it is useful to split it into the entropy relative to the quantum equilibrium $H=-\int \rho \ln (\rho/|\psi|^2) dq$. This relative entropy has been used by Valentini to prove a "subquantum H-theorem" that a general initial distribution will tend toward quantum equilibrium, see for example, http://arxiv.org/abs/1103.1589 for details.
In quantum equilibrium, we have $\rho=|\psi|^2$, so that the formula becomes $H=-\int |\psi|^2 \ln (|\psi|^2) dq$, thus, the standard quantum-mechanical one. After this, you can apply standard quantum theory.