Axionlike particles (ALPs) coupled to nucleons might be copiously emitted from a supernova (SN) core. We extend existing bounds on free-streaming ALPs to the case in which these are so strongly interacting with the nuclear matter to be trapped in the SN core. For strongly interacting ALPs, we also extend the bound from the absence of an ALP-induced signal in Kamiokande-II neutrino detector at the time of SN 1987A. We find that combining the different arguments, SNe exclude values of ALP-nucleon coupling gaN 10-9 for ALP masses ma 1 MeV. Remarkably, in the case of canonical QCD axion models, the SN bounds exclude all values of ma 10-2 eV. This result prevents the possibility for current and future cosmological surveys to detect any signatures due to hot dark matter QCD axion mass.