Adsorption isotherms of alkanes ranging from methane through pentane in the microporous aluminophosphate AlPO4-5 are calculated using grand canonical monte carlo simulations. These simulations predict a surprisingly complex behavior. For methane and ethane we find at low temperatures a low-density-high-density transition that resembles capillary condensation. At intermediate temperatures a high-density structural transition is observed in which the adsorption increases stepwise from four to six molecules per unit cell for methane and two to four molecules per unit cell for ethane. At room temperature a small inflection point in the isotherms signals this transition. For propane, we find a similar high-density transition, but for butane and pentane this transition is not present.