Quintic Modification to Lifshitz Quasi-topological Black Holes

Abstract

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 regular near-horizon expansions together with a numerical 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 remain qualitatively consistent with those found previously in cubic and quartic quasi-topological Lifshitz gravity. A controlled planar comparison through the genuine quartic point \(\hatμ_5=0\) further shows that the very small quintic couplings used in the representative examples produce nearly quartic profiles, whereas larger admissible values of \(\hatμ_5\) lead to clearly resolved changes in both the normalized radial profiles and the Hawking temperature. We also compute the Wald entropy and Hawking temperature and examine the thermal behavior through dimensionless logarithmic temperature--entropy plots. For the representative branches considered here, the heat capacity is positive. For a fixed-theory planar \(z=2\) family, the conserved radial charge is further used to determine the energy density; the differential and integrated first laws are verified numerically to the precision of the computation, and the corresponding Helmholtz free-energy density is found to be negative relative to the Lifshitz vacuum normalization adopted here.