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We analyze the temperature and doping dependence of the specific heat $C(T)$in Na$_x$CoO$_2$. This material was conjectured to undergo a Lifshitz -typetopological transition at $x =x_c =0.62$, in which a new electron Fermi pocketemerges at the $\Gamma$ point, in addition to the existing hole pocket withlarge $k_F$. The data show that near $x =x_c$, the temperature dependence of$C(T)/T$ at low $T$ gets stronger as $x$ approaches $x_c$ from below and thenreverses the trend and changes sign at $x \geq x_c$. We argue that thisbehavior can be quantitatively explained within the spin-fluctuation theory. Weshow that magnetic fluctuations are enhanced near $x_c$ at momenta around $k_F$and their dynamics changes between $x \leq x_c$ and $x >x_c$, when the newpocket forms. We demonstrate that this explains the temperature dependence of$C(T)/T$.  We show that at larger $x$ ($x > 0.65$) the system enters a magnetic quantumcritical regime where $C(T)/T$ roughly scales as $\log T$. This behaviorextends to progressively lower $T$ as $x$ increases towards a magneticinstability at $x \approx 0.75$.

Magnetic fluctuations and specific heat in NaxCoO2 near a Lifshitz topological transition