  Considering a three-dimensional $C$-metric and adding energy-dependent to
this spacetime, we first create a three-dimensional energy-dependent
$C$-metric. Then, we extract accelerating BTZ black hole solutions in gravity's
rainbow. Besides, we show that (A)dS black holes cover by an event horizon that
depends on all the parameters of this theory. Using the definition of Hawking
temperature, we obtain the temperature of these black holes and study the
effects of various parameters on this quantity. We find a critical radius in
which the temperature is always positive (negative) before (after) it. Then, we
obtain the entropy of such black holes. Our analysis indicates that there is
the same behavior for entropy, similar to the temperature. Indeed, before
(after) the critical radius, the entropy is positive (negative). In order to
study the local stability of such black holes, we calculate the heat capacity.
We find two different behaviors for the heat capacity, which depend on the
cosmological energy-dependent constant. As a final result, accelerating AdS BTZ
black holes can satisfy the physical condition and local stability at the same
time.
