Introduction: Carbon futures market plays an important role in energy conservation and emission reduction of energy-intensive firms, and its efficiency is largely reflected by the dynamic relationship between carbon spot and futures prices. Even though the EU carbon futures market has undergone three phases, its efficiency in Phase Ⅱ still contains much controversy.Methods: Based on the cost-of-carry model, we apply linear and nonlinear Granger causality test, vector error correction model and threshold vector error correction model to re-examine the efficiency of the carbon futures market in the second phase of the European Union Emissions Trading System from the perspectives of price discovery and intertemporal arbitrage.Results: i) the EU carbon futures market has the function of price discovery by virtue of the broad scope of participants and the consequent huge liquidity, suggesting that the carbon futures price can predict the carbon spot price; ii) the coefficient requirement of the cost-of-carry model is satisfied and the relationship between the European Union Allowance (EUA) futures and spot prices is consistent with the cost-of-carry model, which indicates that the former is an unbiased estimator of the latter and an equilibrium relationship exists between the two prices so that the carbon futures market is efficient in the long run; iii) however, the carbon futures market shows inefficiency in the short run since it has an error correction function only when the basis is less than −0.07 and the carbon futures price indicates a remarkably slow adjustment to the long-run equilibrium when there is deviation, which can be ascribed to the global financial crisis as well as its aftermath in 2008.Discussion: In the context of global warming and carbon neutrality, the results of this paper are of great significance to emitters, arbitrageurs, hedgers and policymakers in emerging and young carbon markets.