Vortex breakdown is computationally studied by applying an unstable temperature gradient to the axial vortex model known as `Vogel-Escudier' flow. The rotating top plate and the stationary bottom plate are kept at constant but different temperatures. Two main parameters controlling the flow are Reynolds number (Re) and Rayleigh number (Ra). Based on the combinations of these parameters, the flow is either dominated by convection or rotation. The dynamics of the flow in both regimes are distinctively different. For a sophisticated classification of the flow, it is decomposed into the rz - plane and the out-of-the-plane components. The energy content of the two components is analyzed. Using this approach, the flow is categorized in (i) convection dominated, (ii) rotation dominated, and (iii) transitional regime. The analysis of the flow perturbations and thermal plumes reveal rich dynamics exhibited by the flow. The effect of the temperature is also studied on the vortex breakdown bubble and the role of the azimuthal vorticity.