  For $\boldsymbol{a} \in \R_{\geq 0}^{n}$, the Tesler polytope
$\tes_{n}(\boldsymbol{a})$ is the set of upper triangular matrices with
non-negative entries whose hook sum vector is $\ba$. We first give a different
proof of the known fact that for every fixed $\boldsymbol{a}_{0} \in
\mathbb{R}_{>0}^{n}$, all the Tesler polytopes $\tes_{n}(\boldsymbol{a})$ are
deformations of $\tes_{n}(\boldsymbol{a}_{0})$. We then calculate the
deformation cone of $\tes_{n}(\boldsymbol{a}_{0})$. In the process, we also
show that any deformation of $\tes_{n}(\boldsymbol{a}_{0})$ is a translation of
a Tesler polytope. Lastly, we consider a larger family of polytopes called flow
polytopes which contains the family of Tesler polytopes and give a
characterization on which flow polytopes are deformations of
$\tes_{n}(\boldsymbol{a}_{0})$.
