Quintic Modification to Lifshitz Quasi-topological Black Holes

We extend the analysis of Lifshitz black holes to quintic order in five-dimensional quasi-topological gravity coupled to a massive Abelian vector field. Starting from a static ansatz with a constant-curvature horizon, we derive the reduced field equations and identify the radially conserved quantity of the one-dimensional effective system. We then analyze the algebraic conditions that permit Lifshitz backgrounds, both in the absence and in the presence of the massive vector field. Since closed-form black-hole solutions are not available for the generic quintic theory, we construct numerical solutions using near-horizon expansions and a shooting method. We present solutions for the relativistic branch $z=1$ and the Lifshitz branch $z=2$, covering the three horizon topologies $k=-1,0,+1$. The numerical profiles of the metric functions and the gauge-field function show behavior that is qualitatively consistent with earlier studies of cubic and quartic quasi-topological Lifshitz black holes. We also compute the Wald entropy and Hawking temperature, and examine the local thermal behavior through logarithmic entropy – temperature plots. For the representative parameter choices considered here, the numerical branches shown possess positive heat capacity.