To our knowledge, the pricing of climate and green bonds has generally relied upon Markov processes. Contradictorily, recent studies have revealed patterns of mean reversion in temperature data in the United States since 1901 (Liu and Turvey, 2022). We are interested in the application of these findings to the pricing of carbon and climate bonds. We first assess the variance properties of GEO and NGEO carbon futures contracts over time. Both reasonably follow a geometric Brownian motion. We then refer to studies on climatic volatility using historical data from the contiguous United States from 1901 to 2020. Long-range studies of variance in temperature data indicate ergodicity in temperature variation. Our financial analysis links fixed-income products to simulations of carbon futures contracts and temperature based on our findings related to the underlying stochastic processes of both systems. First, we structure carbon-linked bonds that may incentivize farmland managers to adopt less carbon-intensive practices. We find that bond valuation is highly dependent on the observed volatility in each respective futures market. Second, we apply ergodic climate theory to topics in climate finance by modeling 30-year temperature changes with different speeds of mean reversion. We find that catastrophe bonds under more ergodic climatic conditions are valued higher than bonds under less ergodic conditions. These findings suggest that climatic variability is a significant variable that bond issuers must take into account when developing climate-linked, catastrophe-style bonds.