  The Minimal theory of Massive Gravity (MTMG) is endowed non-linearly with
only two tensor modes in the gravity sector which acquire a non-zero mass. On a
homogeneous and isotropic background the theory is known to possess two
branches: the self-accelerating branch with a phenomenology in cosmology which,
except for the mass of the tensor modes, exactly matches the one of
$\Lambda$CDM; and the normal branch which instead shows deviation from General
Relativity in terms of both background and linear perturbations dynamics. For
the latter branch we study using several early and late times data sets the
constraints on today's value of the graviton mass $\mu_{0}$, finding that
$(\mu_{0}/H_{0})^{2}=0.119_{-0.098}^{+0.12}$ at $68\%$ CL, which in turn gives
an upper bound at $95\%$ CL as $\mu_{0}<8.4\times10^{-34}$ eV. This corresponds
to the strongest bound on the mass of the graviton for the normal branch of
MTMG.
