We report a new equation of state (EoS) of cold and hot hyperonic matter constructed in the framework of the quark-meson-coupling (QMC-A) model. The QMC-A EoS yields results compatible with available nuclear physics constraints and astrophysical observations. It covers the range of temperatures from T=0 to 100 MeV, entropies per particle S/A between 0 and 6, lepton fractions from Y$_L$=0.0 to 0.6, and baryon number densities n$_B$=0.05-1.2 fm$^{-3}$. Applications of the QMC-A EoS are made to cold neutron stars (NS) and to hot proto-neutron stars (PNS) in two scenarios, (i) lepton rich matter with trapped neutrinos and (ii) deleptonized chemically equilibrated matter. We find that the QMC-A model predicts hyperons in amounts growing with increasing temperature and density, thus suggesting not only their presence in PNS but also, most likely, in NS merger remnants. The nucleon-hyperon phase transition is studied through the adiabatic index and the speed of sound c$_s$. It is shown that the lowering of (c$_s$/c)$^2$ to and below the conformal limit of 1/3 is a general consequence of instabilities due to any phase transition and is not a unique fingerprint of the hadron-quark matter transition. Rigid rotation of cold and hot stars, their moments of inertia and Kepler frequencies are also explored. The QMC-A model results are compared with two relativistic models, the chiral mean field model (CMF), and the generalized relativistic density functional with hyperons (GRDF-Y). Similarities and differences are discussed.