A search for new particles that decay into top quark pairs is reported. The search is performed with the ATLAS experiment at the LHC using an integrated luminosity of 20.3 fb −1 of proton-proton collision data collected at a centre-of-mass energy of √s = 8 TeV. The lepton-plus-jets final state is used, where the top pair decays to W + b W − b ¯, with one W boson decaying leptonically and the other hadronically. The invariant mass spectrum of top quark pairs is examined for local excesses or deficits that are inconsistent with the Standard Model predictions. No evidence for a top quark pair resonance is found, and 95% confidence-level limits on the production rate are determined for massive states in benchmark models. The upper limits on the cross-section times branching ratio of a narrow Z ′ boson decaying to top pairs range from 4.2 pb to 0.03 pb for resonance masses from 0.4 TeV to 3.0 TeV. A narrow leptophobic topcolour Z ′ boson with mass below 1.8 TeV is excluded. Upper limits are set on the cross-section times branching ratio for a broad colour-octet resonance with Γ /m = 15% decaying to t t ¯. These range from 4.8 pb to 0.03 pb for masses from 0.4 TeV to 3.0 TeV. A Kaluza-Klein excitation of the gluon in a Randall-Sundrum model is excluded for masses below 2.2 TeV.