  We study the hybridizable discontinuous Galerkin (HDG) method for the spatial
discretization of time fractional diffusion models with Caputo derivative of
order $0<\alpha<1$. For each time $t \in [0,T]$, the HDG approximations are
taken to be piecewise polynomials of degree $k\ge0$ on the spatial
domain~$\Omega$, the approximations to the exact solution $u$ in the
$L_\infty\bigr(0,T;L_2(\Omega)\bigr)$-norm and to $\nabla u$ in the
$L_\infty\bigr(0,T;{\bf L}_2(\Omega)\bigr)$-norm are proven to converge with
the rate $h^{k+1}$ provided that $u$ is sufficiently regular, where $h$ is the
maximum diameter of the elements of the mesh. Moreover, for $k\ge1$, we obtain
a superconvergence result which allows us to compute, in an elementwise manner,
a new approximation for $u$ converging with a rate $h^{k+2}$ (ignoring the
logarithmic factor), for quasi-uniform spatial meshes. Numerical experiments
validating the theoretical results are displayed.
