We consider finite temperature effects in a non-standard Bose-Hubbard model for an exciton- polariton Josephson junction (JJ) that is characterised by complicated potential energy landscapes (PEL) consisting of sets of barriers and wells. We show that the transition between thermal activation (classical) and tunneling (quantum) regimes exhibits universal features of the first and second order phase transition (PT) depending on the PEL for two polariton condensates that might be described as transition from the thermal to the quantum annealing regime. In the presence of dissipation the relative phase of two condensates exhibits non-equilibrium PT from the quantum regime characterized by efficient tunneling of polaritons to the regime of permanent Josephson or Rabi oscillations, where the tunneling is suppressed, respectively. This analysis paves the way for the application of coupled polariton condensates for the realisation of a quantum annealing algorithm in presently experimentally accessible semiconductor microcavities possessing high (105 and more) Q-factors.