  We use determinantal quantum Monte Carlo simulations and numerical
linked-cluster expansions to study thermodynamic properties and short-range
spin correlations of fermions in the honeycomb lattice. We find that, at half
filling and finite temperatures, nearest-neighbor spin correlations can be
stronger in this lattice than in the square lattice, even in regimes where the
ground state in the former is a semimetal or a spin liquid. The honeycomb
lattice also exhibits a more pronounced anomalous region in the double
occupancy that leads to stronger adiabatic cooling than in the square lattice.
We discuss the implications of these findings for optical lattice experiments.
