X-ray pulse profile modeling of PSR J0740+6620, the most massive known pulsar, by the NICER and XMM-Newton observatories recently led to a measurement of its radius. We investigate this measurement’s implications for the neutron star equation of state, employing a nonparametric EoS model based on Gaussian processes and combining information from other X-ray, radio and gravitational-wave observations of neutron stars. Our analysis mildly disfavors equations of state that support a disconnected hybrid star branch in the mass-radius relation, a proxy for strong phase transitions, with a Bayes factor of 6.4. For EoSs with multiple stable branches, the transition mass from the hadronic to the hybrid branch is constrained to lie outside ($1, 2$) $M_{\odot}$. We also find that the conformal sound-speed bound is violated inside neutron star cores, implying that core matter is strongly interacting. The squared sound speed reaches a maximum of $0.79^{+0.21}{-0.26}c^2$ at $3.51^{+2.30}{-1.76}$ times nuclear saturation density at 90% credibility. Since all but the gravitational-wave observations prefer a relatively stiff EoS, PSR J0740+6620’s central density is only $3.0^{+1.6}_{-1.6}$ times nuclear saturation, limiting the density range probed by observations of cold, nonrotating neutron stars in $\beta$-equilibrium.