Measurements are made of differential cross-sections of highly boosted pair-produced top quarks as a function of top-quark and $tt¯$ system kinematic observables using proton--proton collisions at a center-of-mass energy of $s=13$ TeV. The data set corresponds to an integrated luminosity of $36.1$ fb$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$, recorded in 2015 and 2016 with the ATLAS detector at the CERN Large Hadron Collider. Events with two large-radius jets in the final state, one with transverse momentum $pT>500$ GeV and a second with $pT>350$ GeV, are used for the measurement. The top-quark candidates are separated from the multijet background using jet substructure information and association with a $b$-tagged jet. The measured spectra are corrected for detector effects to a particle-level fiducial phase space and a parton-level limited phase space, and are compared to several Monte Carlo simulations by means of calculated $\chi 2$ values. The cross-section for $tt¯$ production in the fiducial phase-space region is $292\pm 7 (stat)\pm 76(syst)$ fb, to be compared to the theoretical prediction of $384\pm 36$ fb.