This supplemental material gives some details on the results obtained on the specific heat, CV (T), and the magnetic susceptibility, χ(T), of the antiferromagnetic Heisenberg model on the kagome lattice, using high temperature series expansions (HTSE) and an extrapolating scheme assuming S = 1/2-non gapped as well as gapped low temperature physics, named the entropy method and denoted HTSE+s(e). Various perturbations are explored: impurities, magnetic field, Dzyaloshinskii-Moriya interaction, Ising interaction, second and third neighbor interactions. For all the models used here, new HTSE have been calculated or a few more terms have been added to existing series. Convergence is studied in detail through curves for the raw series in β = 1/T and their Padé approximants (PAs), of the raw series in the energy e (obtained from a Legendre transformation of the previous ones) and their PAs, and of the HTSE+s(e) results. Methods to self-consistently extract the necessary input parameters of the HTSE+s(e) method, i.e. the ground state energy e0 and the T = 0 magnetic susceptibility χ0, are discussed.