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The specific heat $C$ of the single-layer cuprate superconductorHgBa$_2$CuO$_{4 + \delta}$ was measured in an underdoped crystal with $T_{\rmc} = 72$ K at temperatures down to $2$ K in magnetic fields up to $35$ T, afield large enough to suppress superconductivity at that doping ($p \simeq0.09$). In the normal state at $H = 35$ T, a residual linear term of magnitude$\gamma = 12 \pm 2$ mJ/K$^2$mol is observed in $C/T$ as $T \to 0$, a directmeasure of the electronic density of states. This high value of $\gamma$ hastwo major implications. First, it is significantly larger than the valuemeasured in overdoped cuprates outside the pseudogap phase ($p >p^\star$), suchas La$_{2-x}$Sr$_x$CuO$_4$ and Tl$_2$Ba$_2$CuO$_{6 + \delta}$ at $p \simeq0.3$, where $\gamma \simeq 7$ mJ/K$^2$mol. Given that the pseudogap causes aloss of density of states, and assuming that HgBa$_2$CuO$_{4 + \delta}$ has thesame $\gamma$ value as other cuprates at $p \simeq 0.3$, this implies that$\gamma$ in HgBa$_2$CuO$_{4 + \delta}$ must peak between $p \simeq 0.09$ and $p\simeq 0.3$, namely at (or near) the critical doping $p^\star$ where thepseudogap phase is expected to end ($p^\star\simeq 0.2$). Secondly, the high$\gamma$ value implies that the Fermi surface must consist of more than thesingle electron-like pocket detected by quantum oscillations in HgBa$_2$CuO$_{4+ \delta}$ at $p \simeq 0.09$, whose effective mass $m^\star= 2.7\times m_0$yields only $\gamma = 4.0$ mJ/K$^2$mol. This missing mass imposes a revision ofthe current scenario for how pseudogap and charge order respectively transformand reconstruct the Fermi surface of cuprates.

High density of states in the pseudogap phase of the cuprate superconductor HgBa$_2$CuO$_{4 + \delta}$