  Measurements of time delays between multiple quasar images produced by strong
lensing are reaching a sensitivity that makes them a promising cosmological
probe. Future surveys will provide significantly more measurements, reaching
unprecedented depth in redshift, making strong lensing time delay (SLTD)
observations competitive with other background probes. We forecast constraints
on the nature of dark energy from upcoming SLTD surveys, simulating future
catalogues with different numbers of lenses distributed up to redshift $z\sim
1$ and focusing on cosmological parameters such as the Hubble constant $H_0$
and parametrisations of the dark energy equation of state. We also explore the
impact of our ability to precisely model the lens mass profile and its
environment, on the forecasted constraints. We find that in the most optimistic
cases, SLTD will constrain $H_0$ at the level of $\sim 0.1\%$, while the CPL
equation of state parameters, $w_0$ and $w_a$, can be determined with errors
$\sigma_{w_0}\sim 0.05$ and $\sigma_{w_a}\sim 0.3$, respectively. Furthermore,
we investigate the bias introduced when a wrong cosmological model is assumed
for the analysis. We find that the value of $H_0$ could be biased up to $10
\sigma$, assuming a perfect knowledge of the lens profile, when a $\Lambda$CDM
model is used to analyse data that really belong to a $w$CDM cosmology with
$w=-0.9$. Based on these findings, we identify a consistency check of the
assumed cosmological model in future SLTD surveys, by splitting the dataset in
several redshift bins. Depending on the characteristics of the survey, this
could provide a smoking gun for dark energy.
