The relationship between the entanglement creation within coherently pumped and closely spaced two-level emitters longitudinally coupled with a single-mode boson field, and the subsequent quantum cooling of the boson mode are investigated. Even though the two-level qubits are resonantly driven, we have demonstrated an efficient cooling mechanism well below limits imposed by the thermal background. Furthermore, the cooling effect is accompanied by entanglement of the qubit pair components when the dipole-dipole frequency shift is close to the frequency of the boson mode. The maximum boson-mode cooling efficiency is realized at the expense of the entanglement creation. Importantly, this occurs for rather weak external pumping fields protecting the sample from the deterioration. Finally, the conditions to effectively optimize these effects are described as well.