The invisible axion is a candidate for dark matter and is related to the strong CP problem. However, it presents a new cosmological constraint. The axion's energy density from oscillations of the classical axion field does not dissipate rapidly, leading to a potential overclosure of the universe unless the axion decay constant $ f_a $ is less than $ 10^{12} $ GeV. If $ f_a $ is around $ 10^{12} $ GeV, the axion could constitute dark matter. The axion is a pseudo-Goldstone boson with couplings suppressed by $ f_a $. Its mass and couplings are inversely proportional to $ f_a $. The axion field's potential has multiple minima, but in the new inflationary cosmology, the field can be nearly constant throughout the universe. The axion energy density is calculated to be $ \rho_a \sim 10^7 (f_a/m_p)(200 \text{ MeV}/T_i)(A_i/f_a)^2 h^{-2} $, leading to the constraint $ f_a \lesssim 4 \times 10^{12} $ GeV to avoid overclosure. This constraint is more stringent than previous astrophysical limits. The axion energy density is also affected by particle production, but the cosmological redshift prevents significant depletion. The axion field remains coherent today with a small amplitude, and the Peccei-Quinn mechanism still works to solve the strong CP problem. The axion's decay constant must be in the range $ 10^9 $ GeV to $ 10^{12} $ GeV to satisfy both astrophysical and cosmological constraints. If $ f_a \sim 10^{12} $ GeV, axions could make up a significant portion of dark matter. The axion's potential energy density is $ \rho_a \sim f_a^2 m_a^2 $, and it behaves like dust, contributing to the universe's matter content. The axion's energy density fluctuations could seed galaxy formation. The results are consistent with previous studies, and the axion's properties are constrained by both experimental and cosmological observations.The invisible axion is a candidate for dark matter and is related to the strong CP problem. However, it presents a new cosmological constraint. The axion's energy density from oscillations of the classical axion field does not dissipate rapidly, leading to a potential overclosure of the universe unless the axion decay constant $ f_a $ is less than $ 10^{12} $ GeV. If $ f_a $ is around $ 10^{12} $ GeV, the axion could constitute dark matter. The axion is a pseudo-Goldstone boson with couplings suppressed by $ f_a $. Its mass and couplings are inversely proportional to $ f_a $. The axion field's potential has multiple minima, but in the new inflationary cosmology, the field can be nearly constant throughout the universe. The axion energy density is calculated to be $ \rho_a \sim 10^7 (f_a/m_p)(200 \text{ MeV}/T_i)(A_i/f_a)^2 h^{-2} $, leading to the constraint $ f_a \lesssim 4 \times 10^{12} $ GeV to avoid overclosure. This constraint is more stringent than previous astrophysical limits. The axion energy density is also affected by particle production, but the cosmological redshift prevents significant depletion. The axion field remains coherent today with a small amplitude, and the Peccei-Quinn mechanism still works to solve the strong CP problem. The axion's decay constant must be in the range $ 10^9 $ GeV to $ 10^{12} $ GeV to satisfy both astrophysical and cosmological constraints. If $ f_a \sim 10^{12} $ GeV, axions could make up a significant portion of dark matter. The axion's potential energy density is $ \rho_a \sim f_a^2 m_a^2 $, and it behaves like dust, contributing to the universe's matter content. The axion's energy density fluctuations could seed galaxy formation. The results are consistent with previous studies, and the axion's properties are constrained by both experimental and cosmological observations.