  In this work, we make a detailed analysis on the penguin-dominant processes
$B_{(s)}^0 \to \phi\phi \to (K^+K^-)(K^+K^-)$ in the perturbative QCD (PQCD)
approach. In addition to the dominant $P$-wave resonance, the scalar background
$f_0(980) \to K^+K^-$ is also accounted for. We improve the Gegenbauer moments
in $KK$ two-meson distribution amplitudes by fitting the PQCD factorization
formulas to measured branching ratios of three-body and four-body $B$ decays.
We extract the branching ratios of two-body $B_{(s)}^0 \to \phi\phi$ decays
from the corresponding four-body decay modes and calculate the relevant
polarization fractions together with two relative phases
$\phi_{\parallel,\perp}$, which are consistent with the previous theoretical
predictions. The PQCD predictions for the "true" triple product asymmetries
(TPAs) are zero which are expected in the standard model due to the vanishing
weak phase difference, and support the current data reported by the CDF and
LHCb Collaborations. A large "fake" TPA $\mathcal{A}_\text{T-fake}^1=30.4\%$ of
the decay $B^0_s \to \phi\phi \to (K^+K^-)(K^+K^-)$ is predicted for the first
time, which indicates the presence of the significant final-state interactions.
The TPAs of the rare decay channel $B^0 \to \phi\phi\to (K^+K^-)(K^+K^-)$ are
also predicted and can be tested in the near future.
