[Ethereum] accessing global variables cost gas

An approach to solve this is to define a struct inside your library and pass that struct as the first parameter. The functions in your library will be able to modify the struct.

pragma solidity ^0.4.0;

library Arithmetic {
    struct Data {
        bytes1[10] mData;
    }

    function doSomeMath(Data s, bytes1 _input, uint8 _idx) internal pure returns (bytes1) {
        s.mData[_idx] = _input;
        return _input;
    }
}

In your contract you create an instance of that struct and pass it to the library functions. You can use the "using" syntax sugar to have a better looking code.

contract Ballot {
    // Reserver storage for library data
    Arithmetic.Data d;

    // When referencing Data allow shortcut to Arithmetic functions
    using Arithmetic for Arithmetic.Data;

    function bar(bytes1 a) public view returns (uint8) {
        // The following is equivalente to 
        // bytes1 b = Arithmetic.doSomeMath(d, a, 3);
        bytes1 b = d.doSomeMath(a, 3);
        return uint8(b);
    }
}

Here is more information about storing data in solidity https://solidity.readthedocs.io/en/latest/introduction-to-smart-contracts.html#storage-memory-and-the-stack

Those variables, which you defined, will be stored in stack(if there is a space and you keep them in one function). in case if compile will optimize it properly it will be for free, if not, each variable will cost you PUST and POP operations (3+2 gas)

UPDATE

Optimized bytecode costs same amount of gas for both versions. Checked on pragma solidity 0.4.24

Ropsten test (Remix)

pragma solidity ^0.4.24;
contract A {
    uint t;

    function run() public returns(uint){
        uint startTime = 9;
        uint allowedTime = 7;

        uint v = startTime + allowedTime; // second test without v
        t = now + v;
        return t;
    }
}

Results:

with v
  deploy 95237 gas
  func 41470 gas

without v
  deploy 95237 gas
  func 41470 gas