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This work presents an investigation of the Lifshitz black hole thermodynamicsin ($3+1$)-dimensions within $\kappa$-space. We equipped the kinematic term ofHorndeski gravity with the $\kappa$-derivative and through the holographicrenormalization, compute the canonical ensemble and extract the free energy,which carries the $\kappa$-term together with the Horndeski parameters. Theblack hole solution obtained through the no-hair condition establishes arelation between the $\kappa$ and the Horndeski parameters ($\alpha$ and$\gamma$), which provides a Lifshitz black hole without a constraining criticalexponent.

Black holes in asymptotically Lifshitz spacetimes with arbitrary exponents in $\kappa$-Horndeski gravity