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It is known that the linearized Einstein's equation around the pure $AdS$ canbe obtained from the constraint $ \Delta S = \Delta\left< H \right> $, known asthe first law of entanglement, on the boundary $CFT$. The corresponding dualstate in the boundary $CFT$ is the vacuum state around which the linearperturbation is taken. In this paper we revisit this question, in the contextof $ {AdS}_3/{CFT}_2 $, with the state of the boundary ${CFT}_2$ as a thermalstate. The corresponding dual geometry is a planar BTZ black hole. Byconsidering the linearized perturbation around this black brane we show thatEinstein's equation follows from the first law of entanglement. The modularHamiltonian in a thermal state of the ${CFT}_2$ that we have used has beenrecently found in arXiv:1608.01283 [cond-mat.stat-mech].

Linearized Einstein's Equation around pure BTZ from Entanglement Thermodynamics