I'm not sure quite what you mean - life like Earth's can't occur until conditions are suitable to support life like Earth's (the micro-anthropic principle ). On cosmological timescales, the universe had to expand and cool enough for complex structures like stars and galaxies to form. The first few generations of stars would be low in 'metals' (elements above hydrogen & helium), so there would be too little material for many planets and too little carbon and heavier elements around for carbon-based life. Life on Earth seems to have appeared almost as soon as it was capable of supporting life (though not as we know it today), which suggests that if the conditions are right there's a good chance of life appearing.The LCDM cosmology model still violates the Copernicus principle by suggesting that we're living in a privileged position with respect to time.
Most of the rest of the life of the universe is projected to be populated with black holes evaporating until the indefinite heat death period. Compared to that timescale, there's a relatively small window of opportunity early in the life of the universe that is suitable for the emergence of complexity - it's no surprise that we find ourselves in that window.
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