In this paper we study scenarios of the super-Eddington accretion onto black holes at high redshifts z > 10, which are expected to be seeds to evolve to supermassive black holes until redshift z ∼ 7. Such an accretion disk inevitably emitted high-energy photons which had heated the cosmological plasma of the inter-galactic medium continuously from high redshifts. In this case, cosmic history of cosmological gas temperature is modified, by which the absorption feature of the cosmological 21 cm lines are suppressed. By comparing theoretical predictions of the 21cm line absorption with the observational data, we can obtain a cosmological upper bound on the mass-accretion rate as a function of the seed BH masses. In order to realize MBH ∼ 10 9 M at z ∼ 7 by a continuous mass-accretion on to a seed BH, to be consistent with the cosmological 21cm line absorption at z ∼ 17, we obtained an upper bound on the initial mass of the seed BH to be MBH,ini 10 2 M (MBH,ini 10 6 M ) for a seed BH with its comoving number density n seed,0 ∼ 10 −3 Mpc −3 (n seed,0 ∼ 10 −7 Mpc −3 ).