  The ground state of the $S=1/2$ 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 (HTSE) to extrapolate the specific heat $c_V(T)$
and the magnetic susceptibility $\chi(T)$ over the full temperature range,
using an improved entropy method with a self-determination of the ground state
energy per site $e_0$. Optimized algorithms give the HTSE coefficients up to
unprecedented orders (20 in $1/T$) and as exact functions of the magnetic
field. Three extrapolations are presented for different low-$T$ behaviors of
$c_V$: exponential (for a gapped system), 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 $\chi(T=0)$, which could be non zero, from $c_V$ measurements
under different magnetic fields.
