  The transition of a standard thin disk to a radiatively inefficient accretion
flow (RIAF) is expected to occur, when its dimensionless accretion rate is
lower than the critical value. The RIAF is very hot, and it radiates mostly in
the hard X-ray band (>100 keV). Assuming that the accretion disk in every
bright active galactic nucleus (AGN) will finally undergo a RIAF phase while
the accretion rate is lower than the critical value, we calculate the
contribution of the RIAFs in AGNs to the cosmological X-ray background of
10-1000 keV. We find that the timescale of the RIAF accreting at slightly below
the critical rate should be shorter than 0.01 of the bright AGN lifetime if the
critical accretion rate is 0.01, i.e., the accretion rate declines from the
critical value to a rate significantly lower than this critical rate within
this RIAF timescale. The derived RIAF timescale is affected by the parameters
adopted in the model calculations, which is also discussed in this Letter.
