We consider Lifshitz criticalities with dynamical exponent z = 2 that emerge in a class of topological chains. There, such a criticality plays a fundamental role in describing transitions between symmetry-enriched conformal field theories (CFTs). We report that, at such critical points in one spatial dimension, the finite-size correction to the energy scales with system size, L, as ∼ L −2 , with universal and anomalously large coefficient. The behavior originates from the specific dispersion around the Fermi surface, ε ∝ ±k 2 . We also show that the entanglement entropy exhibits at the criticality a non-logarithmic dependence on l/L, where l is the length of the sub-system. In the limit of l L, the maximally-entangled ground state has the entropy, S(l/L) = S0 + 2n(l/L) log(l/L). Here S0 is some non-universal entropy originating from short-range correlations, and n is a half-integer or integer depending on the degrees of freedom in the model. We show that the novel entanglement originates from the long-range correlation mediated by a zero mode in the low energy sector. The work paves the way to study finite-size effects and entanglement entropy around Lifshitz criticalities and offers an insight into transitions between symmetry-enriched criticalities.