For a conditional distribution $p(\theta |y)$, if I specify a one-to-one transformation $\phi = g(\theta)$, how does the Jacobian look like?

Question

Suppose I have a conditional distribution, $p(\theta \mid y)$, where $y$ follows some distribution. This is typically taken from a posterior distribution in Bayesian statistics. If I specify a one-to-one transformation $\phi = g(\theta)$, how does the Jacobian look like? Would it be:

$$ p(g(\theta)\mid y) = p(\theta|y)|g(\theta)|^{-1} $$ ?