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Learning the Hamiltonian underlying a quantum many-body system in thermal equilibrium is a fundamental task in quantum learning theory and experimental sciences. To learn the Gibbs state of local Hamiltonians at any inverse temperature β, the state-of-the-art provable algorithms fall short of the optimal sample and computational complexity, in sharp contrast with the locality and simplicity in the classical cases. In this work, we present a learning algorithm that learns each local term of a n-qubit D-dimensional Hamiltonian to an additive error ϵ with sample complexity $$\tilde{O}\left(\frac{e^{\mathrm{poly}(\beta)}}{\beta^2\epsilon^2}\right)\log(n)$$. The protocol uses parallelizable local quantum measurements that act within bounded regions of the lattice and near-linear-time classical post-processing. Thus, our complexity is near optimal with respect to n, ϵ and is polynomially tight with respect to β. We also give a learning algorithm for Hamiltonians with bounded interaction degree with sample and time complexities of similar scaling on n but worse on β, ϵ. At the heart of our algorithm is the interplay between locality, the Kubo-Martin-Schwinger condition, and the operator Fourier transform at arbitrary temperatures.