Deformed Jaynes–Cummings model (JCM) has a physical importance in quantum optics. Hence, we investigated the nonlinear JCM including the intensity-dependent coupling constant and the additional Kerr term. The cavity was assumed to be in thermal equilibrium with a heat reservoir at temperature T . Using the generators of a closed algebra which reduces to the su(1, 1) and Heisenberg–Weyl algebras at limiting cases, and considering the total excitation number as a constant of motion, the total Hilbert space decomposes into two subspaces. So the eigenvalues and the corresponding eigenvectors were obtained. We derived the thermal density matrix and analysed the fidelity and thermal entanglement using negativity measure. Furthermore, we studied the Berry phase of the nonlinear atom–field system and explored the influence of nonlinearity on the quantum phase transition (QPT) point and entanglement. It is found that the deformation parameter can strongly affect the fidelity, negativity and QPT point.