Normal state specific heat in the cuprate superconductors <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>La</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mo>?</mml:mo><mml:mi>x</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>Sr</mml:mi><mml:mi>x</mml:mi></mml:msub><mml:msub><mml:mi>CuO</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:mrow></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>Bi</mml:mi><mml:mrow><mml:mn>2…

The specific heat $C$ of the cuprate superconductors La$_{2-x}$Sr$_x$CuO$_4$ and Bi$_{2+y}$Sr$_{2-x-y}$La$_x$CuO$_{6+\delta}$ was measured at low temperature (down to $0.5~{\rm K}$), for dopings $p$ close $p^\star$, critical doping onset pseudogap phase. A magnetic field up $35~{\rm T}$ applied suppress superconductivity, giving direct access normal state temperature, enabling a determination $C_e$, electronic contribution normal-state heat, $T \to 0$. In $x=p = 0.22$, $0.24$ $0.25$, $C_e / T 15-16~{\rm mJmol}^{-1}{\rm K}^{-2}$ 2~{\rm K}$, values that are twice as large those higher ($p > 0.3$) lower < 0.15$). This confirms presence broad peak in dependence $C_e$ $p^\star\simeq 0.19$, previously reported samples which superconductivity destroyed by Zn impurities. Moreover, three dopings, we find logarithmic growth 0$, such \sim {\rm B}\ln(T_0/T)$. vs $T$ two typical thermodynamic signatures quantum criticality. very different Bi$_{2+y}$Sr$_{2-x-y}$La$_x$CuO$_{6+\delta}$, again B}\ln(T_0/T$) $p \simeq p^\star$, strong evidence this $\ln(1/T)$ - first discovered cuprates La$_{1.8-x}$Eu$_{0.2}$Sr$_x$CuO$_4$ La$_{1.6-x}$Nd$_{0.4}$Sr$_x$CuO$_4$ is universal property point. All four materials display similar $\rm B$ coefficient, indicating they all belong same universality class.