  Measuring the specific heat of herbertsmithite single crystals in high
magnetic fields (up to $34$ T) allows us to isolate the low-temperature kagome
contribution while shifting away extrinsic Schottky-like contributions. The
kagome contribution follows an original power law $C_{p}(T\rightarrow0)\propto
T^{\alpha}$ with $\alpha\sim1.5$ and is found field-independent between $28$
and $34$ T for temperatures $1\leq T\leq4$ K. These are serious constrains when
it comes to replication using low-temperature extrapolations of
high-temperature series expansions. We manage to reproduce the experimental
observations if about $10$ % of the kagome sites do not contribute. Between $0$
and $34$ T, the computed specific heat has a minute field dependence then
supporting an algebraic temperature dependence in zero field, typical of a
critical spin liquid ground state. The need for an effective dilution of the
kagome planes is discussed and is likely linked to the presence of copper ions
on the interplane zinc sites. At very low temperatures and moderate fields, we
also report some small field-induced anomalies in the total specific heat and
start to elaborate a phase diagram.
