  Gamma-ray bursts (GRBs), as a possible probe to extend the Hubble diagram to
high redshifts, have attracted much attention recently. In this paper, we
select two samples of GRBs that have a plateau phase in X-ray afterglow. One is
short GRBs with plateau phases dominated by magnetic dipole (MD) radiations.
The other is long GRBs with gravitational-wave (GW) dominated plateau phases.
These GRBs can be well standardized using the correlation between the plateau
luminosity $L_0$ and the end time of plateau $t_b$. The so-called circularity
problem is mitigated by using the observational Hubble parameter data and
Gaussian process method. The calibrated \ltb ~correlations are also used to
constrain $\Lambda$CDM and $w(z)$ = $w_{0}$ models. Combining the MD-LGRBs
sample from Wang et al. (2021) and the MD-SGRBs sample, we find $\Omega_{m} =
0.33_{-0.09}^{+0.06}$ and $\Omega_{\Lambda}$ = $1.06_{-0.34}^{+0.15}$ excluding
systematic uncertainties in the nonflat $\Lambda$CDM model. Adding type Ia
supernovae from Pantheon sample, the best-fitting results are $w_{0}$ =
$-1.11_{-0.15}^{+0.11}$ and $\Omega_{m}$ = $0.34_{-0.04}^{+0.05}$ in the
$w=w_0$ model. These results are in agreement with the $\Lambda$CDM model. Our
result supports that selection of GRBs from the same physical mechanism is
crucial for cosmological purposes.
