  We consider the evolution of primordial black holes by including
non-stationarity in their formation process and accretion of radiation in
Brans-Dicke theory. Specifically, we focus on how $\eta$, the fraction of the
horizon mass the black hole comprises capturing nonstationarity, affects the
lifetimes of these primordial black holes. Our calculation reveals that the
primordial black hole dynamics is controlled by the product $f\eta$ where $f$
is the accretion efficiency. We also estimate the impact of $\eta$ through
$f\eta$ on the primordial black holes' initial mass fraction constraint
obtained from the $\gamma$-ray background limit.
