  For steady-state autonomous absorption refrigerators operating in the linear
response regime, we show that there exists a hierarchy between the relative
fluctuation of currents for cold, hot, and work terminals. Our proof requires
the Onsager's reciprocity relation along with the refrigeration condition that
sets the direction of the mean currents for each terminal. As a consequence,
the universal bounds on the mean cooling power, obtained following the
thermodynamic uncertainty relations, receive a hierarchy. Interestingly, within
this hierarchy, the tightest bound is given in terms of the work current
fluctuation. Furthermore, the relative uncertainty hierarchy hands over
additional bounds that can be tighter than the bounds obtained from the
thermodynamic uncertainty relations. Interestingly, all of these bounds
saturate in the tight-coupling limit. We test the validity of our results for
two paradigmatic absorption refrigerator models: (i) a four-level working fluid
and (ii) a two-level working fluid, operating, respectively, in the weak
(additive) and in the strong (multiplicative) system-bath interaction regime.
