The ground state of the kagome Heisenberg antiferromagnet is now recognized as a spin liquid, but its precise nature remains unsettled, even if more and more clues point towards a gapless spin liquid. We use high-temperature series expansions (HTSEs) to extrapolate the specific heat and the magnetic susceptibility over the full temperature range, using an improved entropy method with a self-determination of the ground-state energy per site . Optimized algorithms give HTSE coefficients up to unprecedented orders (20 in ) and as exact functions of the magnetic field. Three extrapolations are presented for different low- behaviors of : exponential (for a gapped system), and linear or quadratic (for two different types of gapless spin liquids). We study the effects of various perturbations to the Heisenberg Hamiltonian: Ising anisotropy, Dzyaloshinskii-Moriya interactions, second- and third-neighbor interactions, and randomly distributed magnetic vacancies. We propose an experimental determination of , which could be nonzero, from measurements under different magnetic fields.