  We present a preliminary lattice calculation of the $D \to \pi$ and $D \to K$
tensor form factors $f_T(q^2)$ as a function of the squared 4-momentum transfer
$q^2$. ETMC recently computed the vector and scalar form factors $f_+(q^2)$ and
$f_0(q^2)$ describing $D \to \pi(K) \ell \nu$ semileptonic decays analyzing the
vector current and the scalar density. The study of the weak tensor current,
which is directly related to the tensor form factor, completes the set of
hadronic matrix element regulating the transition between these two
pseudoscalar mesons within and beyond the Standard Model where a non-zero
tensor coupling is possible. Our analysis is based on the gauge configurations
produced by the European Twisted Mass Collaboration with $N_f = 2 + 1 + 1$
flavors of dynamical quarks. We simulated at three different values of the
lattice spacing and with pion masses as small as 210 MeV and with the valence
heavy quark in the mass range from $\simeq 0.7\, m_c$ to $\simeq 1.2\, m_c$.
The matrix element of the tensor current are determined for a plethora of
kinematical conditions in which parent and child mesons are either moving or at
rest. As for the vector and scalar form factors, Lorentz symmetry breaking due
to hypercubic effects is clearly observed in the data. We will present
preliminary results on the removal of such hypercubic lattice effects.
