GDP increases are generally measured in real (i.e. inflation adjusted) terms, so a doubling of rent costs don't imply a larger GDP. If we're talking nominal GDP, then I'd agree we should expect gov expense to increase as well.

Inflation refers to the money supply and is one number applied to everything in the economy. It does not account for things getting legitimately more expensive. Rent can go up faster than inflation, indeed we'd expect it to as presumably buildings are built or improved to higher standards and the surrounding neighborhoods have improvements which make the local real estate more valuable. Likewise for many other goods and services.