We investigate the thermodynamic cooperativity of folding for a minimalist hydrophobic protein model with an effective reduction in lattice coordination upon chain compaction. This scheme is intended to mimic the requirement that polar backbone groups of real proteins must form hydrogen bonds concomitantly to their burial inside the apolar protein core. In addition to the square lattice, with z=3 conformations per monomer, we use extensions in which diagonal step vectors are allowed, resulting in z=5 and z=7. Thermodynamics are governed by the hydrophobic energy function, according to which hydrophobic monomers tend to make contacts unspecifically while the reverse is true for hydrophilic monomers, with the additional restriction that only contacts between monomers adopting one of z_h < z local conformations contribute to the energy, where zh is the number
of local conformations assumed to be compatible with hydrogen bond formation. Simulations with different native structures were performed and resulted in large increases on the folding transition abruptness and van't Hoff-to-calorimetric-enthalpy ratio. The observed increase in folding cooperativity is correlated to an increase in the convexity of the underlying microcanonical conformational entropy as a function of energy. Simulations in three dimensions, even though using a smaller relative reduction in effective lattice coordination z_h/z=4/5, display a slight increase in cooperativity for a hydrophobic model of 40 monomers and a more pronounced increase in cooperativity for a native-centric Go-model with the same native structure, suggesting that hydrogen bonds are of fundamental importance in the theoretical description of folding cooperativity.