A measurement of the time-dependent CP-violating asymmetry in $Bs0\to \varphi \varphi$ decays is presented. Using a sample of proton-proton collision data corresponding to an integrated luminosity of $5.0$fb$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ collected by the $LHCb$ experiment at centre-of-mass energies $s=7$ TeV in 2011, 8 TeV in 2012 and 13 TeV in 2015 and 2016, a signal yield of around 9000 $Bs0\to \varphi \varphi$ decays is obtained. The CP-violating phase $\varphi sss¯s$ is measured to be $-0.073\pm 0.115$(stat)$\pm 0.027$(syst) rad, under the assumption it is independent on the helicity of the $\varphi \varphi$ decay. In addition, the CP-violating phases of the transverse polarisations under the assumption of CP conservation of the longitudinal phase are measured. The helicity-independent direct CP-violation parameter is also measured, and is found to be $|\lambda|=0.99 \pm 0.05 $(stat)$ \pm 0.01 $(syst). In addition, $T$-odd triple-product asymmetries are measured. The results obtained are consistent with the hypothesis of CP conservation in $b\to s¯ss¯$ transitions. Finally, a limit on the branching fraction of the $B0\to \varphi \varphi$ decay is determined to be at 90% confidence level.