most authors define the canonical form of the exponential family as
$$ p(\mathbf{x} | \boldsymbol{\theta})=h(\mathbf{x}) \exp (\boldsymbol{\eta}(\boldsymbol{\theta}) \cdot \mathbf{T}(\mathbf{x})-A(\boldsymbol{\theta})) $$
with the restriction that $h(x)$ must be non-negative. Why does one not simply define instead
$$ p(\mathbf{x} | \boldsymbol{\theta})= \exp (g(\mathbf{x}) + \boldsymbol{\eta}(\boldsymbol{\theta}) \cdot \mathbf{T}(\mathbf{x})-A(\boldsymbol{\theta})) $$
with $g(\mathbf{x}) = \log h(\mathbf x)$, allowing us to drop the non-negativity assumption since it is sufficed automatically. This also lets us interpret $g(\mathbf{x})$ as a bias term added to the inner product $\langle\boldsymbol{\eta}(\boldsymbol{\theta}), \mathbf{T}(\mathbf{x})\rangle$