This isn't an answer, but it's a bit long for a comment. I am going to write $\varphi$ for your $T$, just for consistency with the notation in a book I'm going to mention.
Let $X = \{ g \hspace{.2pc} \mathrm{d}x^1\wedge \dots \wedge \mathrm{d}x^D: g\in L^p(\mathbb{R}^D)\}$. Then each suitable$_1$ diffeomorphism $\varphi$ of $\mathbb{R}^D$, gives us an associated pullback operator $\varphi^*:X \to X$ which, assuming you stick to orientation-preserving $\varphi$, is just the linear map $g\mapsto f_T$. The quantity you're interested in is then
$$ F[\varphi] = \Vert (I-\varphi^*)g\Vert, $$
where $I:X\to X$ is the identity operator. You then have:
$$ \begin{array}{lll} F[\varphi_1] & = & \Vert (I-\varphi_1^*)g\Vert \\ &=& \Vert ( I-\varphi_2^* + \varphi_2^* - \varphi_1^*)g\Vert \\ &\leq& F[\varphi_2] + \Vert (\varphi_2^* - \varphi_1^*)g\Vert \end{array} $$
whence $F[\varphi_1] - F[\varphi_2] \leq \Vert g \Vert\Vert \varphi_1^*-\varphi_2^*\Vert_\mathrm{op}$ for any pair of orientation-preserving $\varphi_1$ and $\varphi_2$.
I am not what's available in the literature concerning bounds for the operator norm of $\varphi_1^*- \varphi_2^*$, but you might want to take a look at The Pullback Equation for Differential Forms by Csato, Dacorogna and Kneuss in case there are any regularity results in there you can use.