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Abstract We consider top-quark pair production at the LHC within the MS ¯ $$ \overline{\mathrm{MS}} $$ scheme for the renormalisation of the top-quark mass, and we present predictions for total and differential cross sections at next-to-next-to-leading order (NNLO) in QCD. Our state-of- the-art calculation extends the available differential results by one order in the perturbative expansion, and it is relevant for a precise determination of the top-quark mass, including possible effects of the running mass in the MS ¯ $$ \overline{\mathrm{MS}} $$ scheme. We consider variations of the scale at which the MS ¯ $$ \overline{\mathrm{MS}} $$ mass of the top quark is evaluated, extending the usual 7-point to a 15-point scale variation. This additional variation is crucial for a reliable estimate of the theoretical uncertainties, especially at low perturbative orders and close to the production threshold of the top-quark pair. We also compute, for the first time, the invariant-mass distribution of the top-quark pair by using a running mass, evaluated at a dynamic scale. Our predictions for the invariant-mass distribution in the MS ¯ $$ \overline{\mathrm{MS}} $$ scheme are compared with a recent measurement performed by the CMS Collaboration. We observe that the inclusion of the NNLO corrections improves the agreement with the data, and we discuss effects due to the QCD running of the MS ¯ $$ \overline{\mathrm{MS}} $$ mass of the top quark.