Deformations of Lifshitz holography with the Gauss-Bonnet term in ($n+1$) dimensions

We investigate deformations of Gauss-Bonnet-Lifshitz holography in $(n+1)$ dimensional spacetime. Marginally relevant operators are dynamically generated by a momentum scale $\Lambda \sim 0$ and correspond to slightly deformed Gauss-Bonnet-Lifshitz spacetimes via a holographic picture. To admit (non-trivial) sub-leading orders of the asymptotic solution for the marginal mode, we find that the value of the dynamical critical exponent $z$ is restricted by $z= n-1-2(n-2) \tilde{\alpha}$, where $\tilde{\alpha}$ is the (rescaled) Gauss-Bonnet coupling constant. The sub-leading order terms correspond to the marginally relevant modes at the asymptotic region. The generic black hole solution, which is characterized by the horizon flux of the vector field and $\tilde{\alpha}$, is obtained in the bulk, and we explore its thermodynamic properties for various values of $n$ and $\tilde{\alpha}$. We obtain the contribution of the marginally relevant mode and find a linear function $\eta(\tilde{\alpha})$ that relates the free energy density and the energy density when the marginally relevant mode is turned off ($\Lambda =0$).