The Hubble constant has a long history of measurements. It is also one of the hardest cosmological parameters to measure since high accuracy is required.
Thus, at the moment the best guess we have (also supported by a large
set of indirect clues):
We cannot directly measure the age of the universe, but we can measure the age of various astronomical objects. Hopefully, a mother is older than a daughter.
The age of the Earth and the solar system can be measured by means of radioactive dating: 4.6 billion years.
The age of white dwarfs can be measured rather accurately by studying their cooling. The oldest white dwarfs are 10 billion years.
The age of stars can also be determined using theoretical models. The oldest stars are found to be about 13 billion years. This ``measurement'' however is partly based on theoretical modeling, and thus may contain a systematic error.
Thus, the conclusion:
The universe is probably 13 billion years old, but certainly not younger than 10 billion years.