Wentao Liu, Di Wu, Shao-Wen Wei, Yu-Xiao Liu

An exact rotating geometry need not define a black hole in equilibrium when its surface gravity varies across the horizon. We construct an asymptotically flat, non-Kerr subfamily of the aligned null solutions of generalized Proca theory whose outer horizon is exactly Killing and isothermal to second order in rotation. The horizon conditions constrain the vector hair, and integrability of the normalized horizon charge fixes the remaining second-order amplitude. Computing the complete gravitational and nonminimal vector surface charges, we obtain the asymptotically normalized mass and total angular momentum, and derive a logarithmic Hamiltonian entropy. The first law holds in both independent directions of the resulting two-parameter solution space. We express the mass as a function of entropy and angular momentum, recover the geometric thermodynamic potentials, and establish a scale-corrected Smarr relation that retains the dimensionful coupling. The exact metric defines the geometry, while equilibrium thermodynamics is consistently restricted to second order in spin.